ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Handbook of Categorical Algebra 2
Categories and Structures
Prancis/Borceux
Departement de Mathematique
Universite Catholique de Louvain
Wi CAMBRIDGE
'(|P7 UNIVERSITY PRESS

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
1. LA Santalo Integral Geometry and Geometric Probability
2. G E Andrews The theory of partitions
3. R J McEliece The Theory of Information and Coding: A Mathematical Framework for
Comm,unication
4. W Miller, Jr Symmetry and separation of variables
5. D Ruelle Thermodynamic Formalism: The Mathematical Structures of Classical
Equilibrium Statistical Mechanics
6. H Mine Permanents
7. F S Roberts Measurement Theory with Applications to Decisionmaking, Utility, and the
Social Sciences
8. L C Biedenharn and J D Louck Angular Momentum in Quantum Physics: Theory and
Applications
9. L C Biedenharn and J D Louck The Racah-Wigner Algebra in Quantum Theory
10. J D Dollard and C N Friedman Product Integration with Applications in Quantum Theory
11. W B Jones and W J Thron Continued Fractions: Analytic Theory and Applications
12. N F G Martin and J W England Mathematical Theory of Entropy
13. G A Baker, Jr and P Graves-Morris Pade Approximants, Part I, Basic Theory
14. G A Baker, Jr and P Graves-Morris Pade Approximants, Part II, Extensions and
Applications
15. EG Beltrametti and G Cassinelli The Logic of Quantum. Mechanics
16. G D James and A Kerber The Representation Theory of Symmetric Groups
17. M Lothaire Combinatorics on Words
18. HO Fattorini The Cauchy Problem
19. G G Lorentz, K Jetter and S D Riemenschneider Birkhoff Interpolation
20. R Lidl and H Niederreiter Finite Fields
21. W T Tutte Graph Theory
22. J R Bastida Field Extensions and Galois Theory
23. J R Cannon The One-Dimensional Heat Equation
24. S Wagon The Banach-Tarski Paradox
25. A Salomaa Computation and Automata
26. N White (ed) Theory of Matroids
27. N H Bingham, C M Goldie and J L Teugels Regular Variations
28. P P Petrushev and V A Popov Rational Approximation of Real Functions
29. N White (ed) Combinatorial Geometries
30. M Phost and H Zassenhaus Algorithmic Algebraic Number Theory
31. J Aczel and J Dhombres Functional Equations in Several Variables
32. M Kuczma, B Choczewski and R Ger Iterative Functional Equations
33. R V Ambartzumian Factorization Calculus and Geometric Probability
34. G Gripenberg, S-O Londen and O Staffans Volterra Integral and Functional Equations
35. G Gasper and M Rahman Basic Hypergeometric Series
36. E Torgersen Comparison of Statistical Experiments
37. A Neumaier Interval Methods for Systems of Equations
38. N Korneichuk Exact Constants in Approximation Theory
39. R Brualdi and H Ryser Combinatorial Matrix Theory
40. N White (ed) Matroid applications
41. S Sakai Operator Algebras in Dynamical Systems
42. W Hodges Model Theory
43. H Stahl and V Totik General Orthogonal Polynomials
44. R Schneider Convex Bodies
45. G Da Prato and J Zabczyk Stochastic Equations in Infinite Dimensions
46. A Bjorner et al Oriented Matroids
47. G Edgar and L Sucheston Stopping Times and Directed Processes
48. C Sims Computation with Finitely Presented Groups
49. T W Palmer C*-algebras I
50. F Borceux Handbook of Categorical Algebra 1, Basic Category Theory
51. F Borceux Handbook of Categorical Algebra 2, Categories and Structures

Published by the Press Syndicate of the University of Cambridge
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40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, Melbourne 3166, AustraUa
© Cambridge University Press 1994
First published 1994
Printed in Great Britain at the University Press, Cambridge
A catalogue record for this book is available from the British Library
Library of Congress cataloguing in publication data available
ISBN 0 521 44179 X hardback
TAG

d Rene Lavendhomme^
mon maitre

Contents
Preface to volume 2
Introduction to this handbook
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Abelian categories
Zero objects and kernels
Additive categories and biproducts
Additive functors
Abelian categories
Exactness properties of abelian categories
Additivity of abelian categories
Union of subobjects
Exact sequences
Diagram chasing
Some diagram lemmas
Exact functors
Torsion theories
Localizations of abelian categories
The embedding theorem
Exercises
Regul£ir categories
Exactness properties of regular categories
Definition in terms of strong epimorphisms
Exact sequences
Examples
Equivalence relations
Exact categories
An embedding theorem
page xi
XV
1
1
3
8
13
16
21
26
32
34
40
49
51
62
71
86
89
89
92
95
98
101
105
110
Vll

Vlll
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
6
6.1
6.2
Contents
The calculus of relations
Exercises
Algebraic theories
The theory of groups revisited
A glance at universal algebra
A categorical approach to universal algebra
Limits and colimits in algebraic categories
The exactness properties of algebraic categories
The algebraic lattices of subobjects
Algebraic fimctors
Freely generated models
Characterization of algebraic categories
Commutative theories
Tensor product of theories
A glance at Morita theory
Exercises
Monads
Monads and their algebras
Monads and adjunctions
Limits and colimits in categories of algebras
Characterization of monadic categories
The adjoint lifting theorem
Monads with rank
A glance at descent theory
Exercises
Accessible categories
Presentable objects in a category
Locally presentable categories
Accessible categories
Raising the degree of accessibility
Functors with rank
Sketches
Exercises
Enriched category theory
Symmetric monoidal closed categories
Enriched categories
113
120
122
122
125
130
137
139
141
143
146
158
166
173
179
182
186
188
193
197
212
221
231
237
252
254
255
256
263
267
272
277
289
291
292
300

6.3
6.4
6.5
6.6
6.7
6.8
7
7.1
7.2
7.3
7.4
8
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
The enriched Yoneda lemma
Change of base
Tensors and cotensors
Weighted limits
Enriched adjunctions
Exercises
Topological categories
Exponentiable spaces
Compactly generated spaces
Topological functors
Exercises
Fibred categories
Fibrations
Cartesian functors
Fibrations via pseudo-functors
Fibred adjunctions
Completeness of a fibration
Locally small fibrations
Definability
Exercises
Bibliography
Index
Contents ix
309
313
320
325
340
347
349
350
359
366
371
373
374
382
387
394
399
412
424
432
436
439

Preface to volume 2
This second volume of the Handbook of categorical algebra presents a
selection of well-known specialized topics in category theory, with the
exception of toposes which find their natural place in volume 3.
The first great achievement of category theory has certainly been the
theory of abehan categories: these play an important role in homology
and provide the correct setting for studying problems related to exact
sequences. Entire books are devoted to abelian categories; in the first
chapter of this volume we have selected some topics on abelian
categories which appear to remain highly relevant in to-day's research in
general category theory. The chapter starts by estabhshing the key
"exactness" properties of limits and colimits in an abelian category, those
properties being closely related to the existence of an "additive
structure" on the sets of morphisms. The notion of exact sequence is then
introduced together with the technique of "diagram chasing", used to
prove the fundamental diagram lemmas. "Diagram chasing" is a tech-
mque for proving exactness properties in an abelian category, just by
proving them in the categories of modules, where actual elements can
be used. This is best achieved by applying the famous "embedding
theorem" which asserts that every small abehan category can be fully and
®cactly embedded in a category of modules. This very difficult theorem
is proved at the end of the chapter, using a Lubkin completion technique
which turns out to be applicable in many other categorical situations.
For those who do not want to enter these difficult matters, we give a very
dementary "diagram chasing metatheorem" which is good enough for
many purposes. We also study the locaUzations of abelian categories and
their relations with torsion theories and universal closure operations.
Regular and exact categories are in a way those categories which
recapture some essential exactness properties of abehan categories, but
XI

^" Preface to volume 2
without any requirement or implication of additivity. As examples, one
gets most "algebra-like" categories. Regular categories provide the
correct setting for developing the theory of relations, in particular
equivalence relations. Making the technique even harder, the proof we have
given of the embedding theorem for abeUan categories can be adapted
to produce a full exact embedding of every small category in a topos
of presheaves, i.e. in a category Fun(^^P,Set) for some small category
3}. We have preferred the much easier full exact embedding theorem in
a topos of sheaves (see chapter 3, volume 3), which provides almost as
good a "diagram chasing" metatheorem.
The next three chapters are devoted to various categorical approaches
to the notion of "model of an algebraic theory". In chapter 3, we are
interested in those cases where the algebraic structure is given by finitary
everywhere defined operations, the axioms being expressed by equalities.
The corresponding categories of models can be presented as categories
of set-valued finite product preserving functors. We study the
completeness and exactness properties of these "algebraic categories" and pay
special attention to the case of free models. We prove a characterization
theorem for algebraic categories and conclude the chapter with some
special topics: commutative theories, tensor product of theories, and Morita
equivalent theories.
The notion of "monad on a category" formalizes intuitively the idea
of a theory defined by "operations of arbitrary arities". We show the
close relations of this with the notion of adjoint functors and study the
completeness and exactness properties of "monadic categories"; we prove
also a corresponding characterization theorem. We pay some additional
attention to the case of monads with finite rank (intuitively: the
possible arities of operations are now bounded by some cardinal) and we
conclude the chapter by exhibiting some relations with descent theory
for modules.
In chapter 3, algebraic theories were defined using finite products.
In chapter 5, we investigate those theories which can be defined by a
"sketch", that is a set of small limits and colimits. The corresponding
categories of models are the "accessible categories": they turn out to
be exactly the categories of set-valued a-flat functors, for some regular
cardinal a. When just limit cones are used, the corresponding accessible
categories are "locally presentable" and coincide with the categories of
set-valued a-left-exact functors, for some regular cardinal a.
Chapter 6 introduces some fundamental notions and results of
enriched category theory. We are interested here in the case where the

Preface to volume 2 xiii
categories involved have an additional structure on their sets '^{A^B)
of morphisms; for example the categories of modules on a ring have ab-
elian groups of morphisms: they are "enriched" in the category of abelian
groups. We limit our investigations to the basic notions and questions
concerning enriched limits, enriched adjunctions and enriched Kan
extensions. Cartesian closed categories - i.e. those categories in which the
cartesian product with every object has a right adjoint - constitute an
important example of categories in which to enrich category theory: in
particular, all toposes (see chapter 5, volume 3) are cartesian closed.
Unfortunately the category of topological spaces is not cartesian
closed, i.e. given topological spaces y, Z, there is no way to provide the set
C(y, Z) of continuous functions with a topology, in such a way that a
mapping
X >C{Y,Z)
is continuous if and only if the corresponding mapping
XxY >Z
is continuous, for every other space X. We pay some attention to "expo-
nentiable spaces Y"" (those for which the previous problem has a
solution) and show that restricting one's attention to compactly generated
spaces yields a cartesian closed category of topological spaces. Finally we
introduce topological functors, i.e. those functors which satisfy axiomati-
cally the conditions for the existence of topological-like initial structures.
The last chapter of this volume is devoted to the theory of fibred
categories "a la Benabou". Fibred categories formalize the idea of "families
of objects and morphisms" indexed by an object in a base category. We
study first the corresponding fibred notions of adjunction and
completeness. We then pay special attention to "locally small" fibrations, which
are those for which the formal "families of morphisms" are represented
by objects in the base category. We conclude with the very crucial
notion of "definability" which exhibits those classes of devices in a fibration
which can be represented by objects in the base category.

Introduction to this handbook
My concern in writing the three volumes of this Handbook of categorical
algebra has been to propose a directly accessible account of what - in my
opinion - a Ph.D. student should ideally know of category theory before
starting research on one precise topic in this domain. Of course, there are
already many good books on category theory: general accounts of the
state of the art as it was in the late sixties,* or specialized books on more
specific recent topics. If you add to this several famous original papers
not covered by any book and some important but never published works,
you get a mass of material which gives probably a deeper insight in the
field than this Handbook can do. But the great number and the diversity
of those excellent sources just act to convince me that an integrated
presentation of the most relevant aspects of them remains a useful service
to the mathematical conmiunity. This is the objective of these three
volumes.
The first volume presents those basic aspects of category theory which
are present as such in almost every topic of categorical algebra. This
includes the general theory of limits, adjoint functors and Kan
extensions, but also quite sophisticated methods (like categories of fractions
or orthogonal subcategories) for constructing adjoint functors. Special
attention is also devoted to some refinements of the standard notions,
like Cauchy completeness, flat functors, distributors, 2-categories, bicat-
egories, lax-functors, and so on.
The second volume presents a selection of the most famous classes
of "structured categories", with the exception of toposes which appear
in volume 3. The first historical example is that of abelian categories,
which we follow by its natural non-additive generalizations: the regular
and exact categories. Next we study various approaches to "categories
of models of a theory": algebraic categories, monadic categories, locally
XV

xvi Introduction to this handbook
presentable and accessible categories. We introduce also enriched
category theory and devote some attention to topological categories. The
volume ends with the theory of fibred categories "a la Benabou".
The third volume is entirely devoted to the study of categories of
sheaves: sheaves on a space, a locale, a site. This is the opportunity for
developing the essential aspects of the theory of locales and introducing
Grothendieck toposes. We relate this with the algebraic aspects of
volume 2 by proving in this context the existence of a classifying topos for
coherent theories. All these considerations lead naturally to the notion
of an elementary topos. We study quite extensively the internal logic of
toposes, including the law of excluded middle and the axiom of infinity.
We conclude by showing how toposes are a natural context for defining
sheaves.
Besides a technical development of the theory, many people appreciate
historical notes explaining how the ideas appeared and grew. Let me tell
you a story about that.
It was in July, I don't remember the year. I was participating in a
summer meeting on category theory at the Isles of Thorns, in Sussex.
Somebody was actually giving a talk on the history of Eilenberg and
Mac Lane's collaboration in the forties, making clear what the exact
contribution of the two authors was. At some point, somebody in the
audience started to complain about the speaker giving credit to Eilenberg
and Mac Lane for some basic aspect of their work which - he claimed -
they borrowed from somebody else. A very sophisticated and animated
discussion followed, which I was too ignorant to follow properly. The only
things I can remember are the names of the two opponents: the speaker
was Saunders Mac Lane and his opponent was Samuel Eilenberg. I was
not born when they invented category theory. With my little story in
mind, maybe you will forgive me for not having tried to give credit to
anybody for the notions and results presented in this Handbook.
Let me conclude this introduction by thanking the various typists for
their excellent job and my colleagues of the Louvain-la-Neuve category
seminar for the fruitful discussions we had on various points of this
Handbook. I want especially to acknowledge the numerous suggestions
Enrico Vitale has made for improving the quality of my work.

Introduction to this handbook xvii
Handbook of categorical algebra
Contents of the three volumes
Volume 1: Basic category theory
1. The language of categories
2. Limits
3. Adjoint functors
4. Generators and projectives
5. Categories of fractions
6. Flat functors and Cauchy completeness
7. Bicategories and distributors
8. Internal category theory
Volume 2: Categories and structures
1. Abelian categories
2. Regular categories
3. Algebraic categories
4. Monadic categories
5. Accessible categories
6. Enriched categories
7. Topological categories
8. Fibred categories
Volume 3: Categories of sheaves
1. Locales
2. Sheaves
3. Grothendieck toposes
4. The classifying topos
5. Elementary toposes
6. Internal logic of a topos
7. The law of excluded middle
8. The axiom of infinity
9. Sheaves in a topos

1
Abelian categories
1.1 Zero objects and kernels
In section 2.3, volume 1, we studied the notions of terminal and initial
object. In the category of abelian groups, or any category of modules
over a ring, both notions coincide and correspond to the group (or the
module) reduced to {0}.
Definition 1.1.1 By a zero object in a category ^, we mean an object
0 which is both an initial and a terminal object.
It should be noticed that the notion of zero object is autodual!
Definition 1.1.2 Consider a category ^ with a zero object 0. A
morphism f: A >B is called a zero morphism when it factors through the
zero object 0.
Proposition 1.1.3 In a category^ with a zero object 0, there is exactly
one zero morphism from each object A to each object B.
Proof This is just the composite of the unique morphisms A >0,
where 0 is considered as a terminal object, and 0 >B, where 0 is
considered as an initial object. D
Proposition 1.1.4 In a category ^ with a zero object 0, the
composite of a zero morphism with an arbitrary morphism is again a zero
morphism.
Proof Of course, the composite factors through 0. D
Definition 1.1.5 In a category ^ with a zero object 0, the kernel of an
arrow f: A >B is - when it exists - the equalizer of f and the zero
morphism 0: A >B. The cokernel off is defined dually.

2 Abelian categories
Every kernel is, by definition, an equalizer and therefore a monomor-
phism (see 2.4.3, volume 1). But in a category with a zero object, a
monomorphism or even an equalizer need not be a kernel, as is shown
by example 1.1.9.a.
Let us make some trivial observations.
Proposition 1.1.6 Let f be a monomorphism in a category with a zero
object. If f o g = 0 for some morphism g, then g = 0.
Proof f og = 0 = foO {see I.IA), thus g = 0. D
Proposition 1.1.7 In a category with a zero object, the kernel of a
monomorphism f: A >B is just the zero arrow 0 >A.
Proof The composite 0 >A ^ >B is the zero morphism (see 1.1.4)
and if another composite f o g is zero,
X I A 1 >B
0
then fog = / o 0 and therefore p = 0 by 1.1.6, which means that g
factors (uniquely) through the object 0 (see 1.1.3). D
Proposition 1.1.8 In a category with a zero object, the kernel of a
zero morphism 0: A >B is just, up to isomorphism, the identity on
A.
Proof By 1.1.4 0 o 1^ = 0. Now given g: X >A with 0 o p = 0 (of
course!) there exists a unique factorization of g through 1^: it is g itself!
D
Examples 1.1.9
1.1.9.a In the category Gr of all groups and group homomorphisms,
the zero group @) is a zero object and the zero morphisms are precisely
those morphisms which map every element onto the zero element. Given
a group homomorphism /: A >B, its kernel is therefore
Kerf={a€A\f{a) = 0}.
But we have immediately
/(a) = O^WeA f{a' + a-a') = f{a') + /(a) - /(a') = 0
which proves that Ker/ is a normal subgroup of A. Therefore given a
subgroup H C A which is not normal, we have already an example of a
monomorphism (see 1.7.7.C, volume 1) which is not akernel.

1.2 Additive categories and biproducts
H > > A
^ Si
Diagram 1.1
Moreover, applying the amalgamation property for groups, described
in 1.8.5.d, volume 1, we can compute the pushout of diagram 1.1 in Gr
and conclude that 51,52 are injective and the square is a puUback. In
other words, the canonical inclusion i is the equalizer of 51,52, so that
every monomorphism of groups is an equalizer. Finally, not all equalizers
of groups are kernels.
1.1.9.b Let R be a ring and Modi^ the category of right R-modules. The
module @) is clearly a zero object. For a R-linear mapping /: A >B,
it is obvious that
Kerf = {aeA\f{a) = 0},
On the other hand let us consider the submodule f{A) C B and the
corresponding quotient B >B/f{A). It is clear that the composite
A ^ >B 2 >B/f{A)
is zero; but if a composite A ^ >B ^ >C is zero, g maps every element
of f{A) onto 0, thus factors uniquely through p. Therefore
Coker/-S//(^).
Let us observe that given a submodule M C B, M is precisely the
kernel of the projection B >B/M. Moreover, given a surjective linear
mapping p: B >C, C is isomorphic to B/Kerp, thus is the coequalizer
of the inclusion Kerp C B. So every monomorphism is a kernel and every
epimorphism is a cokernel.
1.2 Additive categories and biproducts
Definition 1.2.1 By a preadditive category we mean a category ^
together with an abehan group structure on each set ^{A, B) of mor-

4 Abelian categories
phisms, in such a way that the composition mappings
CABC: nA B) X ^{B, C) >^(A C), {f,g)^gof
are group homomorphisms in each variable. We shall write the group
structure additively
Clearly, the category of abelian groups or, more generally, any
category of modules on a ring is preadditive. As a consequence, every full
subcategory of a category of modules is preadditive as well.
Proposition 1.2.2 The notion of preadditive category is autodual. D
Proposition 1.2.3 In a preadditive category^, the following conditions
are equivalent:
A) ^ has an initial object;
B) ^ has a terminal object;
C) ^ has a zero object.
In that case, the morphisms factoring through the zero object are exactly
the identities for the group structure.
Proof C) implies A), B) and, by duality, it suffices to prove that
A) implies C). Let 0 be an initial object. The set ^@,0) has a single
element, which proves that Iq is the zero element of the group ^@,0).
Given an object C, ^(C, 0) has at least one element: the zero element
of that group. But if /: C >0 is any morphism, / = Iq o / must
be the zero element of ^(C, 0) since Iq is the zero element of ^@,0)
("bilinearity" of the composition). Thus 0 is a terminal object as well.
Given objects C,D € ^^ the groups ^(C, 0) and ^@, jD) are thus
reduced to their zero element, so that the composite of the two zero
elements C >0 >D is the zero element of ^(C, D). D
Proposition 1.2.4 Given two objects A^B in a preadditive category
^, the following conditions are equivalent:
A) the product {P^pa^Pb) ofA,B exists;
B) the coproduct {P^sa^sb) of A,B exists;
C) there exists an object P and morphisms
pa: P >A, pb: P >B, sa: A >P, sb: B >P
with the properties
Paosa = Ia, Pb^sb = Ib, Paosb=0, pbosa = 0,
SA^PA-^ SB OPB = Ip-
Moreover, under these conditions
5A = KerpB , 5B = KerpA , pA = CokersB , pb = Cokersa-

1.2 Additive categories and biproducts 5
Proof By duality, it suffices to prove the equivalence of A) and C).
Given A), define sa- A >P as the unique morphism with the
properties Pa^ SA = ^A^ Pb ^ SA = 0. In the same way sb- B >P is such
that Pa^ SB = 0, Pb ^ SB = Ib' It is now easy to compute that
Pa o {sa ^ Pa -\- SB ^ Pb) = Pa -\- 0 = pa,
Pb o {sa ^ Pa -\- SB ^ Pb) = 0 -\- pb = Pb,
from which sa^Pa-^ sb opB = Ip-
Given condition C), consider C €^ and two morphisms /: C >A,
g: C >B. Define h: C >P as h = sa^ f + sb o g. One has
PAoh=PAOSAof-\-pAOSBog = f-\-0 = f,
PB^h=pBOSAof-\-pBOSBog = 0-\-g = g.
Given /i': C >P with the properties Pa ^ h' = f, pb ^ h' = g, we
deduce
h' = lpoh' = {sa opA-\- sb opb) o h^
= SAopA^h'^-SB^PB^h' = SAof^SB^g
= h.
Now assuming conditions A) to C), let us prove that 5^^ = Kerp^.
We have already Pb ^sa = 0. Choose x: X >P such that pb^x = 0.
The composite pa o x: X >A is the required factorization since the
relations
PAOSAopAOX=pAOX,
Pbosa^Pa^x = Oopj^ox = 0 = pbox
imply sa^pa^x = X. The factorization is unique because pa^sa = 1a
and thus 5^ is a monomorphism.
The relation sb = Kerp^ is true by analogy and the relations pa =
Coker SB, Pb = CokersA hold by duality. D
Definition 1.2.5 Given two objects A^B in a preadditive category a
quintuple {P,Pa,Pb,sa,sb) as in 1.2.4.C) is called a ''biproducf' of A
and B. The object P will generally be written A^ B.
Definition 1.2.6 By an additive category we mean a preadditive
category with a zero object and binary biproducts.
Clearly, the notion of additive category is again autodual. The
following result is somewhat amazing.

6 Abelian categories
Proposition 1.2.7 On a category ^, any two additive structures are
necessarily isomorphic.
Proof Given an object C E ^, we consider the diagonal
Ac: C >CeC
characterized by pi o Ac = Id P2 o Ac = Ic- On the other hand we
consider the difference ac = Pi —P2' C^C >C of the two projections.
It is immediate that
ac o Ac = {pi - P2) o Ac = (pi o Ac) - (p2 o Ac) = Ic - Ic = 0.
We shall prove that gc = Coker Ac-
First of all observe that Ac = Si + S2- Indeed one has
Pi o (si + S2) =piosi = lc =Pio Ac,
P2 o (Si + S2) = P2 o S2 = Ic = P2 o Ac.
Therefore if /: C © C >D is such that / o Ac = 0, one has
/ o si + / o S2 = / o (si + S2) = / o Ac = 0.
Therefore defining g: C >D to be / o si, we have indeed
^ o crc = / o si o (pi - P2) = / o si o pi - / o si o p2
= f osi opi +/0S2OP2 = /o(si opi -\-S20p2)
= /.
Conversely if g': C >D is such that g' ^ cc = f, one has
g^ = {g'olc)-{g'oO) = {g^ opiosi)-{g'op20si) = g^oacosi = fosi.
Thus we have proved that Coker Ac = Pi — P21 which proves that the
difference pi — p2 is characterized, up to isomorphism, by the limit-
colimit structure of ^.
But now given two morphisms a, 6: A ^.C, we have a unique
factorization c: A >C 0 C such that pi o c = a, p2 o c = b. Therefore
a-b={pioc) -{p20c) = {pi - P2) o c,
which proves that the difference a — 6 is characterized by the limit-
colimit properties of ^. And finally a-\-b can be written a — @ — 6),
which concludes the proof. D
If products and coproducts have very special properties in a preaddi-
tive category (see 1.2.4), not much can be said about arbitrary equalizers
and coequalizers. Nevertheless, we have the following.

1.2 Additive categories and biproducts 7
Proposition 1.2,8 Let f^g: A I B be two morphisms in a preaddi-
tive category. The following conditions are equivalent:
A) the equalizer Ker (/, g) exists;
B) the kernel Ker (/ — g) exists;
C) the kernel Ker {g — f) exists.
When this is the case, those three objects are isomorphic.
Proof Since in any case Ker (/, g) = Ker (^, /), it suffices to prove that
A) <=> B). Given a morphism x: X—-^A, fox = goxis equivalent to
(/ ~ 9){^) — 0) from which the result follows. D
Let us conclude this section with a point of notation. If Ai,A2,Bi, B2
are four objects in an additive category ^, a morphism
/: Ai e A2 >Bi e B2
is completely characterized by the four morphisms
/ii =Pio/osi: Ai >J5i,
/12 =Pio f 0S2: A2 >Bi,
/21 =P2o/osi: Ai >B2,
/22 =P2o/oS2: A2 >B2,
so that it makes sense to use the notation
V /21 /22 /
to denote /. It is routine to verify that given another morphism
g: Bi®B2 >Ci®C2,
the composite g o f is precisely represented by the product of the two
individual matrices. That notation extends obviously to the case of n-ary
products. In particular given two morphisms /: Ai >B, g: A2 >B
we write {f,g): Ai 0 A2 >B for the corresponding factorization. In
an analogous way given two morphisms /: A >Bi and g: A >B2,
we write (^): A >Bi 0 B2 for the induced factorization.
Examples 1.2:9
1.2.9.a Every category of modules on a ring is additive.
1.2.9.b The category Gr of all groups and group homomorphisms is
not additive. Indeed, the coproduct of two groups is not their cartesian
product (see 2.2.4.e, volume 1).

8 Abelian categories
1.2.9.C The category Barii of Banach spaces and linear contractions is
not additive. Indeed, isomorphisms are isometric bijections (see 1.9.6.f)
while the product of two objects and their coproduct are not isometric
(see 2.1.7.d and 2.2.4.i, volume 1).
1.2.9.d The category Baricx) of Banach spaces and bounded Unear
mappings is additive: the sum of two bounded linear mappings is again such
a mapping. Notice that binary products and coproducts are computed
as in Barii, but they are now isomorphic since in Barioo isomorphisms
are just the bounded linear bijections (see 1.9.6.e, volume 1).
1.2.9.e The category of abeUan groups is obviously additive. In
particular finite products coincide with finite coproducts (see 1.2.4, volume 1).
But infinite products do not coincide with infinite coproducts (see 2.1.7.C
and 2.2.4.f, volume 1).
1.3 Additive functors
Definition 1.3.1 Given two preadditive categories j/ and ^, a functor
F: si >^ is additive when, for all objects A, A' in j/, the mapping
Faa'-. ^{AA') >^{F{A),F{A')), f^FU)
is a group homomorphism.
It is clear that a composite of additive functors is again additive.
Proposition 1.3.2 Given two preadditive categories s/, ^ with s/
small, the category Add(j/, ^) of additive functors from si to ^ and
natural transformations between them is again preadditive. Moreover
the preadditive structure of Add (j/, ^) is detined pointwise.
Proof Given two natural transformations a : F => G, f3: F => G, define
a -h /?: F =4> G by (a + 0)a = cha + Pa- The details of the proof are
straightforward. D
Proposition 1.3.3 Consider an additive category^ and a small
preadditive category s/. In that case the category Add(j/, ^) is additive and
biproducts in it are computed pointwise. Moreover if ^ is finitely
complete, so is Add(j/, ^) and Rnite limits are computed pointwise.
Proof The constant functor on the zero object of ^ is obviously additive
and it is also the zero object in the category Fun(j/, ^) of all functors
(see 2.15.1, volume 1). Thus it is a zero object in Add(j/,^).
Given two functors F, G: s/ ^^, we consider their pointwise
product F X G: si >0i in the category Fun(j/, Si) of all functors from si to

1.3 Additive functors 9
J^. An object A e <p/ is thus mapped onto F{A)®G{A) and a morphism
/: A >A^ onto the morphism
f Fif) 0 \
I 0 Gif) J
From this description it follows immediately that F x G is an additive
functor and is the biproduct of F and G in Add(j/,^).
If ^ has kernels, consider a natural transformation a: F => G and
its pointwise kernel ^y: K ^ F, let us prove that K is additive. Given
f,g: A I A' in j/ one has
7A' o K{f -9) = F{f -9)o^A = {F{f) - F{g)) o 7^
= {^A'oK{f))-{^A'oK{g))
= ^A'o{KU)-K{g))
from which K{f — g) = K{f) — K{g), since ^a' is a monomorphism. In
the same way, given the zero morphism 0: A >A\
^A'ok{0) = /(O) 07^ = 007^ = 0
from which K{0) = 0 since ^a' is a monomorphism. Thus Add(e5/,^)
has equalizers (see 1.2.8) and we get the conclusion by 2.8.1, volume 1.
D
The next criterion for additivity is particularly interesting.
Proposition 1.3.4 For a functor F: si >^ between additive cate-
gories, the following conditions are equivalent:
A) F is additive;
B) F preserves biproducts;
C) F preserves Gnite products;
D) F preserves finite coproducts.
Proof C) and D) are equivalent by duality C), D) imply in particular
the preservation of both binary products and binary coproducts, thus the
preservation of biproducts (see 1.2.4). To prove the converse implication
B) => C), it remains just to prove the preservation of the product of the
empty family, i.e. of the terminal object 0. For every object B e ^ there
is at least the zero morphism B >F{0), because ^ is preadditive. Now
since 0 E s/ is a. terminal object, the two projections
Pi,P2:oeozz=zi^o

10 Abelian categories
are equal. By assumption, F@ 0 0) = F@) 0 F@) with F{pi) =
F(p2) as projections of the product. Therefore given two morphisms
f,g: B ^F(n) and the factorization (^): B >F(OHF(O) through
the product, the considerations of 1.2.7 show that
thus f = g.So F{0) is the terminal, thus the zero object of ^ (see 1.2.3).
A) ^ B) is obvious and it remains to prove B) ==> A). Assuming
B), we have already seen that F preserves the zero object, thus the
zero morphisms. It remains to prove that F preserves the difference of
two morphisms. Applying again the considerations of 1.2.7, it suffices
to prove that F preserves the difference pi — P2t for every biproduct
A®Am^. Since Fsi^Fs2 are the canonical injections of the biproduct
FA 0 FA^ it suffices to apply the relations
F{pi - P2) o F{si) = F{{pi - P2) o 5i) = F(Ia) = If{A)
{F{pi)-F{p2))oF{si) = lF(A)
and correspondingly with F{s2). □
Let us now make expUcit a canonical example of a category Add(j/, ^).
Consider a ring R with unit. We view it as a preadditive category ^
in the following way:
• ^ has a single object *;
• ^(*, *) = R;
• the composition r o s of two arrows is their product rs as elements
of the ring R;
• the sum r + s of two arrows is their sum as elements of the ring R.
On the other hand consider the (additive) category Ab of abelian groups.
Proposition 1.3.5 Given a ringR with unit, the category Modi? of left
R-modules is isomorphic to the category Add(^, Ab) of additive functors,
where 01 is just the ring R viewed as a preadditive category
Proof Given a left R-module M, define a functor F: M >Ab by the
following data:
• F(*) = (M,+);
• Vr G jR F{r)\ M >M, x y-^ rx.

1.3 Additive functors 11
It is immediate from the axioms of modules that F is an additive functor.
Now given an R-Unear mapping /: M >N to another left R-module
iV, consider the additive functor G associated with N. We get a natural
transformation (p: F =^ G by defining (/?*: M >N to be just /.
Conversely an additive functor F: 31 >Ab gives in particular an
abelian group F(*) and, for every element r E iZ, a group homomorphism
F{t)\ F(*) >i^(*)- Let us provide F(*) with a scalar multiplication
J?xF(*) >i^(*), (r,x) i-^rx = F(r)(x).
The fact that F{r) is a group homomorhism implies
t(x -\- y) = rx -\- ry
and the fact that F is a functor means
{rs)x = r(sx),
Ix = X.
Finally the additivity of F implies
(r + s)x = rx -\- sx,
so that F(*) has been provided with the structure of a left R-module.
If G: 0t > Ab is another additive functor and (/?: F ==> G is a natural
transformation, the group homomorphism (/?*: F(*) >G(*) satisfies
the relation (/?* o F(t) = G(r) o ip^ or, in other words, (p^:{rx) = r(fi^{x)
for every element x € F(*). Therefore (/?* is R-linear.
It is obvious that we have constructed reciprocal isomorphisms. D
Clearly, a right R-module is just an additive functor ^* > Ab, or in
other words a contravariant additive functor ^ >Ab.
Let us finally replace ^ by an arbitrary preadditive category s/. The
additive functors s/ >Ab have many of the properties of ordinary
functors with values in the category of sets. Let us point out three
examples. In fact those examples (as many other results of this section)
are just special cases of a more general theory developed in chapter 6.
Proposition 1.3.6 If j/ is a preadditive category and A e s/, the
"representable functor"
j/(A,-):j/ >Ab, ^{A,-){B) = s/{A,B)
is additive.
Proof Given f,g: X yV in s/ consider
j^iA-)if-g): s^iAX) >^iAY).

12 Abelian categories
One has
^{AJ-g){h) = {f-g)oh = foh-goh = j^{AJ){h)-j^{A,g){h),
from which j/(i4, f — g) = <^{A, f) — s/{A, g). In the same way given
the zero morphism 0: X >F,
j/(A,-)(o)(/i) = oo/i = o. n
Proposition 1.3.7 (Additive Yoneda lemma)
If j^ is a preadditive category, A € <p/ and F: si >Ab is an additive
functor, there exist isomorphisms of abelian groups
Uat{^{A,-),F)^e^^^F{A)
where j^{A,—) stands for the additive representable functor of 1.3.6.
These isomorphisms are natural both in A and in F.
Proof Of course j/(i4, —) is the Ab-valued representable functor
described in 1.3.6.
Given a: j/(yl, —) => F, we define Of,a{(^) = (^a{^a)' Conversely
given a E F{A), we consider for every object B € s/ the mapping
T{a)B: ^{A,B) ^F(B), r{a)B{f) = F{f){a).
Since F is additive, T{a)B is a group homomorphism. Therefore the proof
of 1.3.3, volume 1, applies. D
Proposition 1.3.8 Consider a small preadditive category ^ and an
additive functor F: ^ >Ab. In the category Add(^,Ab) of additive
functors, F can be presented as the colimit of a diagram just constituted
ofAh-valued representable functors and natural transformations between
them.
Proof It suffices to consider the composite
Elts(F) -^ ^ -^ Add (^, Ab)
where Elts(F) is the category of elements of F (see 1.6.3) and ¥""{0) =
^{C^ —). The proof of 2.15.6, volume 1, applies without any change since
colimits in Add(^, Ab) are computed pointwise (see 1.3.3), thus as in
Fun(j?/, Ab) (see 2.15.2, volimie 1), and the additive Yoneda lemma holds
(see 1.3.7). D

1.4 Abelian categories 13
1.4 Abelian categories
Historically, the study of abelian categories played a very important role
in the development of category theory. Algebraic topology and homo-
logical algebra make wide use of them.
Definition 1.4.1 A category ^ is abelian when it satisfies the following
properties:
A) ^ has a zero object;
B) every pair of objects of^ has a product and a coproduct;
C) every arrow of^ has a kernel and a cokernel;
D) every monomorphism of ^ is a kernel; every epimorphism of ^ is a
cokernel
Observing that the dual of each of those four axioms is the axiom
itself, we conclude that
Proposition 1.4.2 (Abelian duality principle)
The dual notion of ^^abelian category" is again ''abelian category". D
The previous proposition is fundamental since it implies that when a
property is proved for abelian categories, so is the dual property.
To provide examples of abelian categories, we now use freely several
results which will be proved in the following sections.
Proposition 1.4.3 If a category ^ is abelian, each localization, and
each colocalization, of^ is again abelian.
Proof An abelian category is finitely complete and finitely cocomplete
(see 1.5.3). By duality (see 1.4.2), it suffices to prove the assertion
concerning locaUzations (see 3.5.5, volume 1). We consider a locaUzation
I H i: S£ ^^; as a full subcategory of a preadditive category, JSf is
obviously preadditive.
If 0 is the zero object of ^, /(O) is both initial and final in JSf (/
preserves colimits by 3.2.2, volume 1, and finite limits by 3.5.5, volume 1)
and 2Z@) is terminal in ^ {i preserves limits by 3.2.2, volume 1); in other
words, viewing i as a full embedding, we conclude that 0 E JSf and 0 is
also the zero object of 5£.
Applying the arguments of 3.5.3,4, volume 1, to the special cases of
those limits and colimits involved in definition 1.3.1, we conclude that
S£ has binary products, binary coproducts, kernels and cokernels.
Now given a monomorphism /: A >B in JSf, it is still a
monomorphism as an arrow of ^ (see 2.9.3, volmne 1). Therefore «(/) = Kerp

14 Abelian categories
with g some arrow of ^, and thus / = li{f) = Ker/(^) since / preserves
kernels.
The case of epimorphisms is more subtle. Consider an epimorphism
/: A >B in JSf. Viewed as an arrow of ^, it has no reason still to be
an epimorphism (and it is not in general). But in ^, / admits a
factorization i{f) = kop where p = Coker (Kerz(/)) and k = Ker (Coker2(/))
(see 1.5.5). Applying / we get / = li{f) = l{k) o l{p) with, in JSf,
/(p) = Coker (Ker/) and l{k) = Ker (Coker/) since / preserves kernels
and cokernels and li{f) is isomorphic to /. But since / is an
epimorphism in JSf, Coker/ = 0 in JSf (see 1.1.7) and therefore Ker (Coker/) is
an isomorphism in JSf (see 1.1.8). Thus / is isomorphic to Coker (Ker/)
in JSf and is a cokernel. D
Proposition 1.4.4 Let ^ be an abelian category and si a small
category. In that case the category of all functors and natural
transformations Fun(j/,^) is again abelian.
Proof By 2.15.1, volume 1, Fun(j/,^) has a zero object, binary
products and coproducts, kernels and cokernels; all those limits and colimits
are computed pointwise.
Choose now a natural transformation a: F ^ G which is a
monomorphism in Fun(j/, ^). The kernel pair of a is just A/?, 1^) (see 2.5.4,
volume 1) and since ^ has kernel pairs (see 1.5.3), the kernel pair of each
morphism a^: F{A) >G{A) is just If{a) (see 2.15.1, volume 1). This
means that each a^ is a monomorphism in ^. Computing the cokernel
0: G =^ H oiain Fun(j/, ^), we know that each /3^: G{A) >H{A) is
the cokernel of the monomorphism a^, thus a^ is the kernel of /3^ (see
1.5.7). This proves that a is the kernel of /3.
The fact that each epimorphism of Fun(j/, ^) is a cokernel follows by
duality. " D
Proposition 1.4.5 Let ^ be an abelian category and si a small
additive category. In that case the category of additive functors and natural
transformations Add(j/,^) is again abelian.
Proof By 1.4.4,1.3.3 and 1.5.3, Add(j/, ^) has finite limits and colimits
computed pointwise as in Fun(j/, ^) (see 2.15.2, volume 1). In particular
it has a zero object, products, coproducts, kernels and cokernels. Now
given a monomorphism a: F => G in Add(j/,^), consider its cokernel
/3: G =^ H in Add(j2/,^). Since kernel pairs and cokernels in Add(«s/,^)
are computed as in Fun(j/,^), a is still a monomorphism in Fun(j2^,^)

1.4 Abelian categories 15
(see 2.5.4) and f3 = Cokera in Fun(e5/,^). Therefore a = Ker/3 in
Fun(j/, ^) and thus a fortiori in Add(j/, ^). D
Examples 1.4.6
1.4.6.a If R is a ring, the category of right (respectively, left) modules
over R admits @) as a zero object and is complete and cocomplete.
Moreover each monomorphism is a kernel and each epimorphism is a
cokernel (see 1.1.9.b). So Modi? is abelian.
1.4.6.b If R is a ring and ^ is a small category, the category of pre-
sheaves of R-modules on ^ (see 4.1.7, volume 3) is again abeUan (see
1.4.4). Indeed, writing ^ for the full subcategory of of Modi? given by
the finitely presented modules (see 5.1.1) and applying 5.2.7, we are
interested in the category Lex(^*, Fun(^*, Set)) of left exact functors from
^* to the category Fun(^*, Set). This category is obviously equivalent
to the category of those functors ^* x ^* >Set which are left exact
in the first variable, i.e. finally to the category Fun(^*, Lex(^*,Set)),
which is just Fun(^*, Modi?) by 5.2.7. Since Modi? is abelian, this last
category is abeUan as well (see 1.4.6).
1.4.6.C If R is a ring and (^, ^) is a site (see 3.2.4, volume 3), we shall
prove that the category of sheaves of R-modules on (^, ^) is a
localization of the corresponding category of presheaves of R-modules on ^,
thus is abelian (see 1.4.6.a and 1.4.3). We write Sh(^,T) for the category
of sheaves on (^, ^); this is a localization of the category Fun(^*, Set)
of presheaves and we write z, a for the canonical inclusion and the
associated sheaf functor (see 3.3.12, volume 3). As in the previous example,
we write ^ for the category of finitely presented R-modules and we are
interested in exhibiting a localization
l-\j: LexG^*,Sh(^,T))lZIZZ;Lex(^*,Fun(^*,Set)).
It suffices to take for j the composite with i, and for / the composite
with a; this makes sense since both i and a are left exact. Applying
3.2.4, volume 1, we get immediately, for F e Lex(^*,Sh(^,T)) and
G€Lex(^*,Fun(^*,Set)),
Nat(Z(G),F) ^ Nat(aoG,F) ^ Nat(G,ioF) ^ Nat(G,j(F)),
from which the adjunction l-\j. I is left exact because finite limits are
C5omputed pointwise both in Fun(^*,Set) and in Sh(^,T) (see 2.15.2,
volume 1, and 3.4.3, volume 3) and are preserved by a.

16 Abelian categories
1.4.6.d We have seen in 1.3.6 that a ring can be seen as a particular
preadditive category ^, while an R-module is just an additive functor
01 >>Ab. More generally if ^ is any small preadditive category, the
category Add(^, Ab) of additive functors is abeUan (see 1.4.5); an additive
functor 0i >Ab is sometimes called a "module on ^".
1.5 Exactness properties of abelian categories
By "exactness properties" we mean essentially properties related with
finite Umits and finite cohmits.
Proposition 1.5.1 In an abelian category, the following conditions are
equivalent:
A) f is an isomorphism;
B) f is both a monomorphism and an epimorphism.
Proof A) =4> B) by 1.9.2, volume 1. Conversely / = Ker^ for some mor-
phism g. But from go f = 0 we deduce ^ = 0 since / is an epimorphism
A.1.6) and therefore / is an isomorphism A.1.8). D
Proposition 1.5.2 In an abelian category ^, the intersection of two
subobjects always exists.
Proof Consider two monomorphisms a: A> >C and b: B> >C.
There exist morphisms /: C >D and g: C >E with a = Ker/,
b = Kerg. Let us consider the factorization (^): C >D x E.
In the case ^ = Ab, one could write
A={c€C\f{c) = 0},
B={ceC\g{c) = 0},
AnB = L€c\ (^^{c) = o\ ,
so that AnB would just be Ker {0. Let us generalize this to the case of
an arbitrary abeUan category ^. We refer to diagram 1.2.
We consider k = Ker {0. Since f o k = pd ^ {g) ^ k = pd ^ 0 = 0, k
factors through a as A: = aoa'; in the same way, k = bob'. Let us prove
that (Ker (^), a', V) is the puUback of the pair (a, b). Given n, v such that
aou = bov, we have fobov = foaou = Qou = 0; since a = Ker/, there is
a unique w such that kow = boy. So boy = kow = bob'ow and b'ow = y
since 6 is a monomorphism. Moreover aou = bov = kow = aoa'ow^
from which u = a' ow since a is a monomorphism. This factorization w
is unique since a, 6 are monomorphisms (see 1.7.2, volume 1). D

1.5 Exactness properties of abelian categories
X
17
•> B
A —> C —> D
E ^
Pe
PD
-Dx E
Diagram 1.2
Proposition 1.5.3 An abelian category is finitely complete and finitely
cocomplete.
Proof By duality, it suflices to prove the finite completeness. Since
finite products exist by definition, it suffices to prove the existence of
equalizers (see 2.8.2, volume 1). We refer to diagram 1.3. Let us consider
two morphisms f^g: A \ R Observe that (^) and i}^) are monomor-
phisms, since their composite with the projection p^'- Ax B >A is
just the identity on A. The intersection (P, tx, v) of those two monomor-
phisms exists by 1.5.2.
In the case of the category of abelian groups, one would have
A S {(a,/(a))|a6A},
A ^{{a,g{a))\a^A],
P ^ {{a,a')\ aeA,a'eA,a = a', f{a) = g{a')}
^ {o|aeyl, fia) = gia)},
and therefore P would be the equalizer of / and g. Let us generalize this
fact to the case of an arbitrary abelian category •<?.
We refer still to diagram 1.3. Composing it with the first projection

18
Abelian categories
A >-
(V)
Diagram 1.3
> A
(V)
^Ax B
PA' a X B >A we obtain
u = 1a o u = Pa o Cf) o u = Pa ^ Cg) ^ V = '^A ^ V = V,
while composing with the projection ps- Ax B >B yields
f ou = pbo (^) ou = pbo (^^) ov = gov.
Putting these equalities together, we get fou = gou. Now if x: X >A
is another morphism such that fox = gox, then clearly C'f)ox = (^^)ox
and we find a unique factorization y such that uoy = x = v oy. D
Proposition 1.5.4 For a morphism f: A >B in an abelian category
^, the following conditions are equivalent:
A) f is a monomorphism;
B) Ker/ = 0;
C) ^C e"^ "ig-.C >A fog = 0^g = 0.
Proof A) :=^ B) and A) ^ C) have been proved in 1.1.7 and 1.1.6. To
prove B) ==> A), we refer to diagram 1.4. If Ker/ = 0, consider two mor-
phisms tx,V such that f ou = f ov and their coequalizer q = Coker(tx,v)
(which exists by 1.5.3). Since q is an epimorphism, q = Coker ly for
some w. Since fou = fov, f factors through q as f = m o q. From
f ow = moqow = moO = 0 we deduce that w factors through the kernel
koifasw = kon. But since Ker f = 0^w = 0 and g is an isomorphism
(see 1.1.8). In particular g is a monomorphism and since qou = qo v^
one concludes that u = v. So / is a monomorphism.
Finally let us prove C) =^ B). Clearly the composite 0 >A ^ >B
is the zero morphism. Now if the composite C ^ >A ^ >B is the zero

1.5 Exactness properties of abelian categories
19
Ker/<..
n
w
f
-> •
m
Diagram 1.4
morphism, g factors through 0 by assumption and this factorization is
unique since 0 is terminal. Thus 0 = Ker /. D
Theorem 1.5.5 Every morphism f in an abelian category can be
factored uniquely (up to isomorphism) as f = i o p, where i is a mono-
morphism and p is an epimorphism. Moreover, i = Ker(Coker/) and
p = Coker(Ker/).
The factorization of / refered to in theorem 1.5.5 is called, for obvious
reasons, the "image factorization" of /. Observe that in the case of a
homomorphism /: A >B of abelian groups, the image f{A) C B is
such that /: A >f{A) is surjective. Therefore f{A) is the quotient of
A by the kernel of /, i.e. the cokernel of the kernel of /.
Proof We refer to diagram 1.5 where the various morphisms are
introduced as follows. First, / is the given morphism, k = Ker/ and
p = CokerA:. Since fok = 0{k = Ker/), / factors throughp as f = iop.
We shall prove first that i is a monomorphism. To do this we apply 1.5.4.
It suffices to choose x such that i o x = 0 and prove that x = 0. Since
iox = 0, i factors uniquely as i = rog, where q = Cokerx. Now qop is an
epimorphism as composite of two epimorphisms, thus there exists some h
such that qop = Coker h (see 1.4.1). Since foh = roqopoh = roO = 0, /i
factors uniquely ash = kol (k = Ker/). Finally poh = pokol = Oo/ = 0,
thus p factors as p = 5 o (g o p) since qop = Coker h. But since p is an
epimorphism this last relation implies soq= 1. So g is a monomorphism
from which qo x = 0 implies x = 0. Thus we have already proved that
/ = 2 o p, with i a monomorphism and p = Coker (Ker /).

20
Abelian categories
-> •
-^ •
Diagram 1.5
■^> • >-
Diagram 1.6
-> •
• —» • > ^ > •
h
-> •
The uniqueness of such a factorization is attested by 4.4.5, volume 1,
since regular epimorphisms are strong (see 4.3.6, volume 1).
By duality, / factors qs f = i' op' with i' = Ker(Coker/) and p^ an
epimorphism. By imiqueness of the mono-epi factorization, the
factorizations f = i o p and f = i' o p' axe isomorphic, which concludes the
proof. n
Proposition 1.5.6 In an abelian category, consider diagram 1.6 where
the outer rectangle is coimnutative, p,q are epimorphisms and i,j are
monomorphisms. There exists a unique morphism h making the
completed diagram conmiutative.
Proof Every epimorphism is regulax (see 1.4.1), thus strong (see 4.3.6,
volume 1), so the result follows from 4.4.5, volume 1. D
The previous construction is generally referred to as the "naturality"
of the image construction.
Proposition 1.5.7 In an abelian category

1.6 Additivity of abelian categories
P
21
■^> /
/h
-> B
in
J >-
Diagram 1.7
A) every monomorphism is the kernel of its cokernel,
B) every epimorphism is the cokernel of its kernel
Proof By 1.5.5, an epimorphism / can be factored as / = iop, where i is
a monomorphism and p = Coker (Ker/). But since / is an epimorphism,
so is i and therefore i is an isomorphism (see 1.5.1); finally / is isomorphic
to p = Coker (Ker/). The first assertion follows by duality. D
Proposition 1.5.8 Consider a morphism a: A >B in an abelian
category, with image factorization a = i o p. Given another factorization
a = uo V, with u a monomorphism, there exists a unique morphism h
such that uoh = i, hop = v (see diagram 1.7).
Proof Apply 1.5.6 with f = v^ q = Ij^ j = u and g = Ib- □
Observe that the previous statement is just a particularization of the
definition of a strong epimorphism p (see 4.3.5, volume 1).
1.6 Additivity of abelian categories
Having in mind the considerations of 1.2.7, we shall now introduce an
additive structure on every abelian category.
Lemma 1.6.1 Consider an object A of an abelian category, the diagonal
morphism A: A >A x A and its cokernel q: A x A >Q. The object
Q is isomorphic to A.
Proof We refer to diagram 1.8 where pi,p2 denote the two projections
of the product Ax A. We define r: A >Q to be the composite qo^^)
and we shall prove it is an isomorphism. Since A is a monomorphism,
A = Ker (Coker A) = Kerg (see 1.5.7). Since pi o (^^) = 1^, pi is an
epimorphism and (^q) is a monomorphism; in the same way p2 is an
epimorphism and (j^) is a monomorphism. By definition p2 o (^^) = 0
and if p2 o V = 0, (^^){piov) = v as checked inmiediately by composing

22
X
l4
Abelian categories
-—> A ^ V
(V)
A ) ^ >AxA ^> Q
t
Diagram 1.8
-> Y
with both projections pi,P2- This factorization pi ot; is unique since (^q)
is a monomorphism, thus {^q) = Kerp2- In the same way d^) = Kerpi.
But since p^ is an epimorphism, p2 = Coker (Kerp2) = Coker {^q). In the
same way pi = Coker d^).
To prove that r* is a monomorphism, choose x such that r o x = 0.
From q o (}^) o X = r o X = 0 and A = Kerg, we deduce a factorization
{^q) ox = Aoy. Therefore y = p2 o A o y = p^ o (^^) o x = 0 o x = 0.
It follows that A o y = (^^) o X = 0 and since (^(f) is a monomorphism,
then X = 0. By 1.5.4, r* is a monomorphism.
We have still to prove that r is an epimorphism (see 1.5.1). Choose z
such that z or = 0. From z o q o (^^) = z o r = 0 and P2 = Coker (^^),
we deduce a factorization z o q = t o p2. Therefore
t = top2oA = zoqoA = zoO = 0.
It follows that zoq = top2 = 0 and thus z = 0 since p2 is an epimorphism.
So r* is an epimorphism as well (see 1.5.4). D
Definition 1.6.2 Consider two arrows /, g: ^ ) A in an abelian
category ^. With the notation of 1.6.1, we write a a for the composite
Ax A 2_
^Q-
>A
and call it "substraction on A". We write f — g for the composite
B—^-^^A X A—^^^-^A
and de&ne f + g = f — {0- g).
We shall now prove that definition 1.6.2 introduces a (pre)additive
structure on the abelian category V, The following lemma will be crucial.

1.6 Additivity of abelian categories
fxf.
^
BxB-
(TB
-^A X A
oa
B
-> A
f
Diagram 1.9
B
f
-» A
f X f
B X B^ ^-^A X A
» B
■?■ > A ^^^— A
J
f
Diagram 1.10
23
Lemimia 1.6.3 Given a morphism f: B >A in an abelian category
and with the notation of 1.6.2, f o ctb = (ta^ {f x /); see diagram 1.9.
Proof We consider diagram 1.10 where the relation
CTA o (/ X /) o Ab = cr^ o Aa o / = 0 o / = 0
implies the existence of a unique g such that goaB = (ta^ {f x f), just
because by construction, as = Coker A^. We must prove that f = g.
With the notation of 1.6.1, we have also q o C^) = r, from which
^A^io) "= r"^ ogo C^) = 1a. In the same way as o Co) = 1b- On the
other hand the relations
P2o(/x/)o (^^^^=0 = p^o (^^^^ of

24
(a
b
c
/a
b
c
\d
Abelian categories
-> {AxA)x{AxA) ^^^^—> Ax A
(^AxA
AxA
Pi
Diagram 1.11
(^AxA
AxA
(^A
Diagram 1.12
(^A
-> A
-> {Ax A) X {Ax A) —''^^''^ ) AxA
(TA
-> A
imply (/ X /) o {\^) = (V) o /. Therefore
g = go(TBO { ^\=(jAo{fxf)oi ^\=(jAoi Mo/ = /. D
Theorem 1.6.4 Every abelian category is additive.
Proof Let us first apply lemma 1.6.3 with B = A x A and / the i-th
(z = 1,2) projection. Given an arbitrary object C and four morphisms
a, 6, c, d: C > A, consideration of diagram 1.11, where z = 1,2, shows
the validity of the formula
(:)-E)=(n)-
Next let us apply lemma 1.6.3 with again B = A x A, but f = cta-
Given an arbitrary object C and four morphisms a, 6, c, d: C > A, the
consideration of diagram 1.12 together with the previous formula, shows
the validity of the relation
(a - c) - F - d) = (a - 6) - (c - d).
On the other hand we have noticed in the proof of 1.6.3 that (Ta^Cq ) =
1a, which implies that given a morphism a: C >A, one has a — 0 = a.
In the same way the relation a a o A^ = 0 implies a — a = 0.

1.6 Additivity of abelian categories 25
It is now rather straightforward to prove the theorem. Applying the
previous relations, one proves successively that
@ - 6) - c = @ - 6) - (c - 0) = @ - c) - F - 0) = @ - c) - 6,
0 - @ - d) = (d - d) - @ - d) = (d - 0) - (d - d) = (d - 0) - 0 = d,
6 + c = 6 - @ - c) = @ - @ - 6)) - @ - c)
= @-0)-(@-6)-c) = @-0)-(@-c)-6)
= @ - @ - c)) - @ - 6) = c - @ - 6) = c + 6,
6 + @ - c) = 6 ~ (@ ~ @ - c)) = 6 - c,
6 + @ - 6) = 6 - 6 = 0,
0 - (c - d) = @ - 0) - (c - d) = @ - c) - @ - d) = @ - c) + d,
0 - (c + d) = 0 - (c - @ - d)) = @ - c) + @ - d) = @ - c) - d,
(a - 6) + d = (a - 6) - @ - d) = (a - 0) - F - d) = a - F - d),
(a + 6) + d = (a - @ - 6)) + d = a - (@ - 6) - d) = a - @ - F + d))
= a + F + d),
which shows already that ^(C, A) has been provided with the structure
of an abeUan group.
Now given an arrow x: X >C,
{a — b) o X = (ta ^ i , ) ox = cr^ol j=aox — 6
ox.
On the other hand given an arrow y: A >Y', we can apply lemma
1.6.3 with f = y, yielding
yo{a-b) = yoaAol ^ j= (TYo{yxy)ol ^ j= ayol ^ ° ^ )= {y^CL)-{y^b).
The conclusion follows at once. D
Corollary 1.6.5 Let A^B be two objects of an abehan category, with
product {A X B,pa,Pb) Sind coproduct {AamalgB, sa, sb)- There exists
an isomorphism Ax B = AUB and, via this isomorphism,
SA = KerpB, SB = KerpA, PA = CokersB, Pb = CokersA,
PAOSa = 1a, PbOSb = 1b, SAOpA-hSBopB = lAeB,
where we have written A^ B to denote the object Ax B = AUB.
Proof See 1.2.4. D

26 Abelian categories
1.7 Union of subobjects
First of all, we deduce from the considerations of 4.2, volume 1:
Proposition 1.7.1 In an abehan category, the intersection and the
union of a finite family of subobjects always exists.
Proof The intersection of two subobjects exists by 1.5.2. The union of
two subobjects exists by 4.2.6, volume 1, because an abelian category
has coproducts (see 1.5.3), epi-mono factorizations (see 1.5.5), and every
epimorphisms is regular (see 1.4.1), thus strong (see 4.3.6, volume 1).
Moreover every object A of an abelian category has a biggest subobject
(i.e. the intersection of the empty family), namely A itself. Now the
unique morphism 0 >A is a monomorphism because 0 is terminal; it
represents the smallest subobject of A because 0 is initial. D
In other words:
Corollary 1.7.2 In an abelian category the subobjects of every object
constitute a lattice, with top and bottom elements. D
In the category of sets, the distributive formulae
are well-known to be vaUd for subsets R> >A, Si> >A of a set A.
Nothing analogous holds for abehan categories, i.e. the lattices of sub-
objects are no longer distributive. For example, in the category Vect^ of
real vector spaces, consider the following subobjects:
si: R> >R2^ ^^. jj^ ^^2^ ^. jg^ ^jg2^
where si and S2 are the two canonical axes and A is the diagonal. Clearly
51 V S2 = R^, si A A = @) and S2 A A = @). Therefore
(si V S2) A A = R2 A A = A,
E1 A A) V E2 A A) = @) V @) = @),
which shows that in general
E1 V 52) A A 7*^ E1 A A) V E2 A A).
But imions in an abelian category share another important property
of imions in the category of sets: namely, eflfectiveness.

1,7 Union of subobjects 27
Rns> > s
R > >RL\S
Diagram 1.13
Definition 1.7.3 Consider a category ^ with binary intersections and
binary unions of subobjects. The union of two subobjects r: R> >A
and s: S> >A is effective when RuS is the pushout of R, S over their
common subobject RnS. In other words, the square in diagram 1.13 is
both a puUback and a pushout.
Rephrasing the previous definition, the union RuS is effective when, in
defining a morphism RuS >B, it suffices to find morphisms R >B,
S >B which agree on RD S. See exercise 1.15.6 for an example of a
non-effective union.
Proposition 1.7.4 In an abelian category binary unions are effective.
Proof By 4.2.6, volume 1, we know that RuS is obtained as the image
factorization
R^S S-^>RUS> ^ >A
of the morphism (r, s): R 0 S >A. By 1.5.5, p = Coker (Ker (r, s)).
Let us first compute Ker (r, s). Giving a morphism x: X >R® S is
equivalent to giving a pair (^^) of morphisms, with xi = Prox: X > J?,
X2 = ps ox: X >S. The relation {r,s)ox = 0 is just (r, s) o (^^) = 0,
or more explicitly (r* o xi) + (s o ^2) = 0, which we can write r o xi =
(—s) 0x2- Observe that —15: S >S is an isomorphism (it is its own
inverse!) so that {S, —s) is still a subobject of A (for a while,
avoiding ambiguity in mentioning subobjects requires writing the monomor-
phisms explicitly, not just their sources). Since intersections of
subobjects are just puUbacks (see 4.2.3, volume 1), giving a pair xi: X > J?,
^2' X >S such that r oxi = (—s) o X2 is equivalent to giving a
morphism x^: X >{R, r) D E, -s). All this proves that the kernel of
(r,s): R^S >A
is exactly (iJ, r) fl E, —s) or, to make notation easier, the morphism
k: RnS >ReS,

28
Abelian categories
Rns-
-» S
R
-> A
Diagram 1.14
where Rr\ S stands now for the iisual intersection oi R,S as in
diagram 1.14 and k is the factorization of the two morphisms
-Is
RnS^
-^R, RnS>-
^s-
^S
->y, /3: S >Y
through the biproduct R^ S.
Now we must prove that for two morphisms a: R-
which agree on Rn S (i.e. a o tx = /3 o v), there exists a common and
unique extension 7: J? U S >Y. Considering the factorization
(a,/3): J?eS'-
^Y
and the relation p = Coker (Ker (r, s)), it suffices to prove the relation
(a, /3) o Ker (r, s) = 0. Using the description of A: = Ker (r, s) we have just
given, we have indeed
{a,f3)ok= (a,/3)o ( ^ j = {aou) - {Cov) = 0.
D
Using the "quotient notation" p: A >A/R for the coequaUzer of a
monomorphism R> >A^ we obtain the following corollary:
Corollary 1.7.5 If r: R> >A and s: S> >A are subobjects in an
abelian category, A/{RU S) is the pushout of A/R and A/S under A.
In other words the square in diagram 1.15 is a pushout.
Proof We refer to diagram 1.16 where r,s,m,nj are the original
monomorphisms, p = Cokerr, q = Cokers and t = Coker/. Prom tor =
tolom = Owe find a factorization u yielding t = uop and in the same
way we find v such that t = voq.
Now consider x and y such that xop = y oq. One has x opol om =
X opor = 0 and xopolon = yoqolon = yoqos = 0, from which
X opol = 0 since m, n are the canonical morphisms of a pushout (see
1.7.4). Therefore we find a factorization z with the property zot = xop.

n
m
R >-
1.7 Union of subobjects
A >A/R
A/S >A/RyjS
Diagram 1.15
RyjS<—'^—< S
-> A
^>A/R
29
Diagram 1.16
This impUes zouop = zot = xop and thus z ou = x, since p is an
epimorphism. In the same way z o v o q = z o t = x o p = y o q and
z ov = y since q is an epimorphism. Since t is an epimorphism, such a
factorization z is necessarily unique. D
Let us now prove a property which generaUzes somewhat the existence
of effective unions (see 1.7.3). It should be observed that proposition
1.7.6 and its dual do not hold in general for arbitrary categories (see
exercise 1.15.7).
Proposition 1.7.6 In an abelian category ^, the pushout of a mono-
morphism is still a monomorphism and the pushout square is also a
pullback. More precisely if the square in diagram 1.17 is a pushout and

30
Abelian categories
f
A > B
C
-> D
Diagram 1.17
if g is a monomorphism, k is a monomorphism as well and the square is
a puUback. By duality, in an abelian category the puUback of an
epimorphism is still an epimorphism and the corresponding puUback square is
also a pushout.
Proof We refer to diagram 1.18. We consider the biproduct B^C and
the morphism
/
( ^)= iBeco ( ^\ = {sB.sc)o ( ^\ = -{sBof) + {scog).
We put p = Coker/, h = po sc-, k = po sb- This yields immediately
(hog)^(kof) = {poscog)-(posBof) =po{{scog)-{sBof)) =pol = 0,
thus hog = ko f. Moreover, if m: B >M, n: C >M are such that
mof = nog^we consider the factorization {m^n): B^C >M, which
yields '
(m,n) ol = (m,n) o
(-/)
-(mo f)^ (nog) = 0,
from which we get a unique factorization d: D >M through the co-
kemel of /, yielding dop = (m,n). This impUes immediately
dok = zopoSB = (m,n) oSB = rn
doh = zoposc = (rn, n) o sc = rn
and the imiqueness of such a factorization follows at once from the
uniqueness conditions in the definitions of (m,n) and d. So the
previous constructions describe the pushout (h, k) of the pair (g, /); compare
with 2.17.1, volume 1.
Let us prove now that I is a monomorphism. Choosing x: X >A
such that Z o x = 0, it suffices indeed to compose on the left with pc to

1.7 Union of subobjects
31
Diagram 1.18
get
0 = pcoO = pcolox=pco ((sc og) - {sBof)) ox = gox,
from which x = 0 since ^ is a monomorphism. Thus / is indeed a
monomorphism (see 1.5.4) and therefore / = Ker(Coker/) by 1.5.7; in other
words, / = Kerp.
Now let us choose y: Y >B such that koy = 0^ asm diagram 1.18.
One has
poSb oy = koy = 0
from which one gets a factorization z: Y >A of y through the kernel
/ of p, yielding lo z = sb oy. Let us observe that
goz=pco
'f
oz=pcoloz=pc^SB^y = Ooy = 0
and thus z = 0, since ^ is a monomorphism. Finally SB^y = l^z = 0
and y = 0 because sb is a monomorphism. By 1.5.4 again, this proves
that A: is a monomorphism.
We must still prove that the pushout square is a puUback. Let us
consider diagram 1.18 again, where now ho x = k oy and q = Cokerg.
Since go^ = 0 = 0o/, we find a unique r such that roh = q, rok = 0.
Therefore qox = rohox = rokoy = Ooy=:0, But since ^ is a

32 Abelian categories
0 > A
SA
B >-^^->AeB
Diagram 1.19
monomorphism, g = Ker(Coker^) = Kerg (see 1.5.7) and thus there is
a z such that g o z = x. Prom koy = hox = hogoz = kofoz, we
deduce y = foz because A: is a monomorphism. Such a ;2; is unique since
^ is a monomorphism. D
We conclude this section with a property of biproducts.
Proposition 1.7.7 Given a biproduct A>^^A 0 B^^^^B, the
intersection of the subobjects sa, sb is the zero object and their union is
AeB,
Proof Given morphisms x: X >A^ y: X >B such that sa^ x =
SB o y, one has x = pa ^ sa ^ x = pA o sb o y = 0 o y = 0. In the same
way 2/ = 0, so that x and y factor (uniquely!) through X >0. So the
square of diagram 1.19 is a puUback, but it is also a pushout since 0 is
initial. Applying 1.7.4, we conclude that A^ B is the union of s^, sb-
a
1.8 Exact sequences
Several aspects of this section are valid for additive or even for pread-
ditive categories. But for the sake of simplicity, we shall restrict our
attention to the cage of abelian categories.
Definition 1.8.1 In an abelian category ^, a composable pair of
morphisms
A ^—>B 2 ^c
is called an exact sequence when the image of f coincides with the kernel
ofg.
To be precise, in 1.8.1 we consider the image factorization
A 2—^>/> ^ >B
of /; the requirement is thus that (/, i) is the kernel of g.

1.8 Exact sequences
33
Diagram 1.20
Proposition 1.8.2 Consider diagram 1.20 in an abelian category ^,
where both triangles are commutative while {i,p) and (j, q) are the image
factorizations of f and g. The following conditions are equivalent:
A) (/, g) is an exact sequence;
B) (i, g) is an exact sequence;
C) (/, q) is an exact sequence;
D) (i, q) is an exact sequence.
Proof It suffices to observe that Ker^ = Kerg, since j is a monomor-
phism. n
Corollary 1.8.3 In abelian categories, the notion of exact sequence is
autodual.
Proof With the notation of 1.8.2, the sequence {f^g) is coexact when,
in the dual category ^*, {g*,f*) is an exact sequence. By 1.8.2 this is
equivalent to q* = Kerz* in ^*, thus to q = Cokerz in ^. But since i is
a monomorphism, i = Ker(Coker2) = Kerg (see 1.5.7) and the sequence
(/,^) is exact. By duality, exactness implies coexactness. D
Definition 1.8.4 A finite or infinite sequence of morphisms
in A Jn+l
-^An-
-^A
n+l"
->An+2-
> • • •
in an abelian category is called exact when each pair of consecutive
morphisms is an exact sequence in the sense of 1.8.1.
Proposition 1.8.5 In an abelian category, the following equivalences
hold:
A) 0 >A ' >B is an exact sequence iff f is a monomorphism;
B) B ^ >A >0 is an exact sequence iff f is an epimorphism;
C) 0 >A ^ >B ^ >C is an exact sequence iff f = Kerg;
D) C ^ >B * >A >0 is an exact sequence ifff = Coker^.

34 Abelian categories
Proof By duaUty, it suffices to prove A) and C). Since 0 is terminal,
each morphism 0 >A is a monomorphism. Thus 0 >A ^ >B is
exact when 0 = Ker/ (see 1.8.1), i.e. when / is a monomorphism (see
1.5.4); this proves A) and impUes immediately C). D
Definition 1.8.6 By a short exact sequence in an abelian category is
meant an exact sequence of the form
0 >A ^ >B 2 >c ^0.
Proposition 1.8.7 Consider a short exact sequence
0 >A ^ >B 2 ^c >0
in an abelian category The following conditions are equivalent:
A) there exists a morphism s: C >B such that go s = Ic;
B) thre exists a morphism r: B >A such that r o / = 1^;
C) there exist morphisms s: C >B, r: B >A such that the
quintuple {B^r^g^f^s) is the biproduct of A and C.
Proof By duality, it suffices to prove A) <^ C). But C) =4> A) is obvious
(see 1.6.5). Given A), consider the morphism 1b — sog\ B >B. One
has ^0A^—80^) = g — gosog = g—g = 0, from which one gets a unique
morphism r: B >A such that f or = 1b — so g^ since / = Ker^. We
have already f or -\- so g = 1^, go f = 0 and g o s = Ic- We have also
forof = {lB-sog)f = f-sogof = f-0 = f.thusrof = 1^ since
/ is a monomorphism, see 1.8.5A). Finally f or o s = {1b — so g)s =
s — sogos = s — s = 0, thus ros = 0 since / is a monomorphism. This
concludes the proof (see 1.2.4). D
Definition 1.8.8 In an abelian category a split exact sequence is a
short exact sequence which satisfies the conditions of 1.8.7.
Let us recall that given a biproduct as in 1.2.4, the sequence
0 >A 5^^—>AeB—2^-^B >0
is always exact since sa = Kerp^? Pb = Cokersa'^ see 1.8.5. It is a split
exact sequence by 1.8.7.3. And 1.8.7 asserts that all split exact sequences
axe of this type up to isomorphism.
1.9 Diagram chasing
In any abelian category Modi^ of modules over a ring iJ, monomorphisms
are just injections, and epimorphisms are just surjections, as attested by
1.7.7.e and 1.8.5.e, volume 1. Therefore

1.9 Diagram chasing 35
/: A >B is a monomorphism
iff \/aeA f{a) = 0 =^ a = 0
iff Va, a' eA f{a) = f{a') ^ a = a',
/: A >B is an epimorphism
iff \/b€B 3a e A f{a) = b,
A ^ >B ^ >C is an exact sequence
iff \/aeA g{f{a)) = 0,
\/b€B g(b) = 0 ^ 3a^Ab = f{a).
In the case of the exact sequence, the first condition says exactly that
Im / C Ker^, while the second condition means Ker^ C Im /.
In the case of a category of modules, proofs concerning exact
sequences, thus in particular monomorphisms, epimorphisms, kernels and
cokernels, can be performed quite easily using the characterization in
terms of elements. This technique is called "diagram chasing". We shall
prove that, amazingly enough, this same technique of "diagram chasing"
can be applied to every abelian category.
The final result concerning diagram chasing in an abelian category is
obtained by proving that every abelian category s/ can be fully
embedded in a category of modules over a ring, in a way which preserves and
reflects exact sequences (see 1.14.9). We shall present here a weakened
version of the diagram chasing theorem, very easy to prove and good
enough for many purposes.
Definition 1.9.1 In an abelian category ^, consider an object A and a
morphism f: A >B.
A) A pseudo-element of A is an arrow • ^ >A with codomain A; we
shall write simply a €* A;
B) two pseudo-elements X ^ > A and X' —^ A are pseudo-equal when
there exist epimorphisms Y—^>X, Y-^-»X' with the property
aop = a' op'; we shall write simply a =* a';
C) the pseudo-image under f: A >B of a pseudo-element • ^ >A of
A is the composite f o a; we shall write simply f(a).
Proposition 1.9.2 Consider two morphisms f: A >B, g: B >C
of an abelian category The following properties hold:
A) pseudo-equality is an equivalence relation on the pseudo-elements of
A;
B) for pseudo-elements a €* A, a' €* A
a=* a' => /(o)=*/(a');

36
Abelian categories
P"
C) for a pseudo-element a E* A
/(ffW)
-*ifog)ia).
Proof The reflexivity and the symmetry of pseudo-equaUty are
obvious; let us prove the transitivity. We consider a^a'^a'' E* A with
a ="" a'^ a' =* a". There exist epimorphisms p^p'^p"^p'" making all
parts of diagram 1.21 commutative; we obtain new epimorphisms q^q'
by performing the puUback of p'^p" (see dual of 1.7.6). The equality
ao{poq) = a" o (p'" o q') allows us to conclude the proof. The validity
of the two other statements is obvious. D
Lemma 1.9.3 Let A he an object of an abelian category. There exists an
equivalence class for pseudo-equality on pseudo-elements of A, consisting
exactly of all the zero morphisms with codomain A.
Proof Consider a: X >A pseudo-equal to 0: Y >A. There exist
epimorphisms p,q such that aop = Oog = 0; thus a = 0. Conversely if
a = 0, it suffices to consider the epimorphisms px- X ^ Y >X and
py: X ®Y >Y (see 1.2.4) to get aop^ = 0 = Oopy. D
The previous lemma allows us to refer freely to "the" zero pseudo-
element of a given object. The interest of the notion of pseudo-element
is attested by the following result:

1.9 Diagram chasing
37
Y
P
-» X
X <^
-» B
Diagram 1.22
P
f
-> B
Proposition 1.9.4 In an abelian category^, the following equivalences
hold:
A) f: A >B is the zero morphism
iff Va E* A f{a) =* 0;
B) f: A >B is a monomorphism
iff Va E* A f{a) =* 0 ^ a =* 0
iff \/a,a' E* A f{a) =* /(a') ^ a =* a';
C) f: A >B is an epimorphism
iff V6 E* B 3a E* A /(a) =* b;
D) A ^ >B ^ >C is an exact sequence
iff Va E* A g{f{a)) =* 0,
V6 E* B g(b) =* 0 -^ 3a E* A /(a) =* 6;
f5j if/: A >B and a,a' E* A with f{a) =* /(a')
then there exists a'' E* A such that /(a") =* 0 and
yg:A >C ^(aO=*0 ^ ^(a") =* ^(a).
Proo/ A) If / = 0, then /(a) = / o a = 0 and one gets the conclusion
by 1.9.3. Conversely, / = /(l^) =* 0 and thus / = 0, again by 1.9.3.
B) If / is a monomorphism and /(a) =* /(a'), there are epimor-
phisms p, q with foaop=foa'oq and thus aop = a' o q. So the
second characterization holds and it implies the first one just by putting
a' = 0. This first characterization is just condition 1.5.4.C) for being a
monomorphism.
C) If / is an epimorphism and b E* S, construct the pullback of /
and 6, getting a E."" A and an epimorphism p, as in diagram 1.22. From
/oaoly = 6opwe deduce /(a) =* b. Conversely since \b ^* B^ we
find a e."" A and epimorphisms p, q such that foaop = q (see the second
square in diagram 1.22). Since g is an epimorphism, so is /.
D) Suppose first that (/, g) is an exact sequence. Since ^ o / = 0,
we already have the first assertion (see first part of the proof). Now
choose b e* B such that g{b) =* 0 and consider the image factorization

38
Abelian categories
Y —» X
-» / >
> B
-> C
Diagram 1.23
> C
Diagram 1.24
/ = iop of / as in diagram 1.23. Since gob = 0, there is a factorization c,
yielding 6 = i o c, through the kernel i = Ker^. Computing the pullback
of (p, c), we get a €* A and an epimorphism q (see 1.7.6). The relation
/oaoly = iopoa = boq implies /(a) =* b.
Conversely suppose condition D) is satisfied and consider again the
image factorization / = z op of /, as in diagram 1.24. We must prove
that i = Kerg. From (a) and the first part of the proof we deduce that
goiop = go f = 0^ thus goi = 0 since p is an epimorphism. Now if b is
such that gob = 0, (b) impUes the existence oia €* A and epimorphisms
^,r such that iopoaor = boq. Computing the pullback (c, j) of (i, 6), we
find a factorization z such that joz = q and coz = poaor. The arrow j
is a monomorphism since i is (see 2.5.3, volume 1) and an epimorphism
since q is (see 1.8.2, volume 1). Thus j is an isomorphism (see 1.5.1) and
6 factors through i as b = io coj~^. The factorization is luiique since i
is a monomorphism.
E) We have the pseudo-elements a: X >A, a'\ X' >A and two
epimorphisms p: Y >X, p'\ Y >X' such that f opoa = f op' oa'.

1.9 Diagram chasing
39
C
f
-> D
Diagram 1.25
It suffices to define a'' = poa — p' o a'. D
The proof of 1.9.4.E) indicates that a" E* A should be thought of,
intuitively, as the difference a — a' E* A. Now let us be careful: this
does not provide the pseudo-elements of A with the structure of an
abeUan group. For example, given a pseudo-element a: X >A of A,
the relation a o (—Ix) = (~a) o Ix implies that a =* —a... while in
general one does not have a -\- a ='^ 0. This same example shows the
weakness of this notion of pseudo-elements: it cannot be used to prove
the equality of two morphisms. In other words given f^g: A ^B
^ae*A f{a)=''g{a)
does not imply f = g. The previous remarks show indeed that putting
g = —f will give (in general) a counterexample.
Here is another (useful) example of a proposition which is just an
implication, not an equivalence. Again this is due to the fact that our
present notion of pseudo-element is too weak.
Proposition 1.9.5 In an abelian category s/ consider the puUback in
diagram 1.25. Given two pseudo-elements c E* C and b E* B such that
f{c) =* g{b), there exists a pseudo-unique pseudo-element a €* A such
that h{a) =* c, k{a) =* b.
Proof If /(c) =* g{b), there are epimorphisms p, q such that f ocop =
g oboq. By definition of a puUback, this implies the existence of some
a^"" A such that hoa = cop^ koa = boq. In particular h{a) =* c and
k{a) =* b.
Consider now a' E* A such that h{a') =* c and k{a^) =* b. There are
epimorphisms p', g',p", g" such that hoa'op' = coq' and koa'op" = boq".
All the epimorphisms p,p^,p^',q,q',q" can, by successive puUbacks, be
replaced by epimorphisms with the same domain, from which a =* a'.
D

40
Abelian categories
0 0
A ^ > B ^ > C
7
D ^^ > E —> F
A)
B)
-> G
-> H
-> /
Diagram 1.26
To get a better insight into what a pseudo-element a: X >A is, let
us consider its image factorization X—^>I>-^A. Prom the relation
a o Ijj^ = i o p we deduce a =* i, so that each pseudo-element can be
represented by a monomorphism. Now if two monomorphisms i: /> >A
and j: J> >A are pseudo-equal, we have epimorphisms g, r* such that
ioq = jf or. The imiqueness of the image factorization (see 1.5.5) implies
the isomorphism of i and j. In conclusion, the equivalence classes of
pseudo-elements of A are in bijection with the subobjects of A (see
4.1.1, volume 1).
1.10 Some diagram lemmas
In this section, we freely use 1.8.5 and 1.9.4 without referring to them
every time.
Lemima 1.10.1 (The kernels' lemma) Consider diagram 1.26 in an
abelian category, with conmiutative squares A) and B) and exact rows
(C>^)> (Oj^jO- P^^ 7 = KerS, 6 = KerA, e = Ker/x. There exist unique
morpbisms a,P making the diagram conunutative. Moreover, {a,P) is
an exact sequence.

7
D
a
fi
1.10 Some diagram lemmas
a
A)
-* B
-^ G
-» B
f3
-» C
-» E
V
-» F
Diagram 1.27
/?
-^ C
7
-» //
Diagram 1.28
-> jD
-> /
-> 0
TT
41
-^ E
-> J
Proof Prom Ao(^o7 = i/o^o7 = i/oO = 0 and 6 = KerA, we deduce
the existence and uniqueness of a. In an analogous way, we construct /3.
Since eoj3oa = 770(^07 = O07 = 0 and e is a monomorphism, /3oa = 0.
Now choose h e* B such that j3{h) =* 0. Since G706)F) =* {eof3){h) =*
e{fi) =* 0, we get some d e* D with the property C,{d) =* 6F). From
{y o ^)(d) =* (A o Q{d) =* (A o 6){h) =* 0, we deduce e{d) =* 0, since
1/ is a monomorphism. Therefore we have a E."" A such that 7(a) =* d.
Then {6 o a){a) =* C7(a) =* C(d) =* 5(&), from which a{a) =* 6 since 6
is a monomorphism. D
Lemma 1.10.2 In an abelian category, consider diagram 1.27, where
the two rows are exact, the square A) is a puUback and the other square
is commutative. Then e is a monomorphism.
Proof Choose c e* C such that e{c) =* 0. Since /3 is an epimorphism,
there exists b e* B such that /3F) =* c. From {rjo6){b) =* {eof3){b) =*
e{c) = 0, we deduce the existence oi d €* D such that C{d) =* 6{b).
Since A) is a puUback, by 1.9.5 there is some a e* A with the properties
7(a) =* d and a{a) =* b. Finally c =* /3F) =* (/3 o a) (a) =* 0. D
Lemma 1.10.3 (The five lemma) In an abelian category, consider
diagram 1.28 which is commutative with exact rows. Ife, C, 0, A are
isomorphisms, Tj is an isomorphism as well.

42
0
^ A
7
-^ D
Abelian categories
-^ S
-» E
Diagram 1.29
-> C
-> F
^ 0
-> 0
Proo/ By duality and proposition 1.5.1, it suffices to prove that rj is
a monomorphism. For this choose c E"^ C such that 7y(c) =* 0. Then
{0 o 7)(c) =* {(, o 7y)(c) =* ^@) =* 0, thus 7(c) =* 0 since ^ is a
monomorphism. Therefore we find h E* B such that /3F) =* c. Since
{y o Q{h) =* (ry o j3){h) =* 7y(c) =* 0, there is / E* F such that /i(/) =*
(^F). Since e is an epimorphism, we can choose a e* A such that e{a) =*
/. {C, o a)(a) =* (/i o e)(a) =* /i(/) =* CF), from which a{a) =* 6 since
C is a monomorphism. Finally c =* /3F) =* (/3 o a)(a) =* 0. D
Lemma 1.10.4 (The short five lemma) In an abelian category
consider diagram 1.29, where both squares are commutative and both rows
are exact. If^ and e are monomorphisms, 6 is a monomorphism as well.
Proof Choose a pseudo-element b E* B such that 6{b) =* 0. From the
relation {e o /3)F) =* (ry o 6){b) =* rj{0) =* 0 we deduce that /3F) =* 0,
since e is a monomorphism. Therefore we find a £* A such that a{a) =*
b. Prom (C o 7) (a) =* {6 o a) (a) =* 6{b) =* 0, we deduce a =* 0 since
both C and 7 are monomorphisms. Finally b =* a(a) =* a@) =* 0. D
Lemma 1.10.5 (The nine lemma) In an abelian category, consider
diagram 1.30, where all squares are commutative. Suppose five rows and
colunms are exact, including the central row and the central column.
Then the remaining row or column is exact as well. Moreover the square
A) is a puUback and the square B) is a pushout.
Proof Just by synmietry, the roles of rows and colunms can be
interchanged. So let us suppose that the three colunms are exact as well
as the central row. The roles of the first and the last row can now be
interchanged by duality, so that we can assume the last row to be exact.
The arrow a is a monomorphism since 7 and C are. Applying lenmnia
1.10.1, it remains to prove that /3 is an epimorphism.
Choose c €* C. Then £{c) E* F and, since 77 is an epimorphism,
e{c) =* r){e) for some ee* E.So {^oX){e) =* {fJiorj){e) =* {fJ'Oe){c) =* 0

0
1.10 Some diagram lemmas
0 0 0
7
-» G
A)
-^ H
B)
-^ A —> B —> C
-> D —> E —> F
-» /
43
-» 0
-» 0
^ 0
Diagram 1.30
and therefore A(e) =* u{g) for some g €* G. Since 9 is an epimorphism,
there exists d e* D such that 0{d) =* g. We have thus (A o C){d) =*
{voO){d) =* v{g) =* A(e). Applying 1.9.4E), there is a pseudo-element
writtten formally e - C,{d) €* E such that A(e - C(d)) =* 0 and, since
G7 o C){d) =* 0, 7?(e - C(d)) =* 7?(e). From A(e - C(d)) =* 0 we deduce
the existence of 6 €* B such that 6{h) =* e - C,{d). Finally (e o ^)F) =*
G7o6)F) =* ■q{e — C,{d)) =* 7?(e) =* e(c) and since e is a monomorphism,
m =* c.
Let us prove now that A) is a puUback; by duality, B) will be a
pushout. Given morphisms x: X >D, y: X >B such that C o x =
Soy, one has eol3oy = r)o6oy = r]o(^ox = 0, and thus l3oy = 0, since
e is a monomorphism. From /Soy = 0 and a = Ker/3 we get z: X ^A
sueh that aoz = y. Then C°7°z = 6oaoz = 6oy = (^ox and 7o2 = x
since C is a monomorphism. Such a 2 is necessarily unique since a and
7 are monomorphisms. D

44
Abelian categories
0 0 0
-> A
^ B >-
B/A
-> C
C/A
-> 0
^>C/B-
C/B
-> 0
-> 0
Diagram 1.31
When /: A> >B is a monomorphism, let us again use the notation
p: B >B/A for the cokernel.
Lemma 1.10.6 (First Noether isomorphism theorem)
In an abelian category, consider subobjects A> >B> >C. In this case
B/A is a subobject of C/A and (C/A)/{B/A) is isomorphic to C/B.
Proof It suffices to apply the nine lemma to diagram 1.31, where the
existence of a last row follows from 1.10.1. D
Lemma 1.10.7 (Second Noether isomorphism theorem)
Consider two subobjects R> >A and S> >A in an abelian category
The following isomorphism holds:
S/{RnS)^{R\JS)/R.
Proof Consider diagram 1.32, where the two first rows and the two first
columns are exact. Since A) is a pullback, the last row is exact as well

1.10 Some diagram lem,m.as
0 0 0
->i?nS'-
-> R
-> S
A)
B)
^R/iRnS)-.iRuS)/s-.^^0/^.
45
■^s/iRnS) > 0
->i?U5 ^{RUS)/R y 0
^ 0
Diagram 1.32
(see 1.10.2). Applying 1.10.1 we construct the last column and lemma
1.10.5 implies its exactness.
Applying 1.10.5 again, we know that B) is a pushout square. But by
1.7.5 the pushout of {R U S)/R and {R U S)/S is just {R U S)/{R U S),
i.e. the zero object. Therefore we have an exact sequence
0
■^{RUS)/R-
>S/{RnS)
which imphes the required isomorphism (see 1.5.1).
^0,
D
Lemma 1.10.8 (The restricted sneike lemma) In an abelian
category, consider diagram 1.33, where all squares are commutative and
where all rows and columns are exact. In those conditions there exists
a diagonal morphism uj which makes the sequence {/3,uj,t) an exact
sequence.
Proof To construct lj, we consider diagram 1.34 where T, A are
obtained as puUback of e, ry and A, S as pushout of n, v. We also define

46
Abelian categories
0 0
■^ K
^
■^ L
0
Diagram 1.33
0
> 0
> 0
■^ 0
S = Ker A and T = CokerS. By lemma 1.10.1 and its dual, there are
morphisms ^ and Q. making the diagram commutative and the two outer
columns exact. Since E = Ker A and A is an epimorphism (dual of 1.7.6),
A = Coker (Ker A) = Coker S (see 1.5.7); in the same way 5 = KerT. So
in diagram 1.34, just the central column is not exact.
Since AoAoToS = ^071060"^ = 0 and A = CokerS, there is a
unique factorization x such that x^^ = AoAoF. Now T o x o A =
ToAoAor = rio/xoeoA = 0, so that T o x = 0 because A is an
epimorphism. Prom T o x = 0 and S = KerT, we obtain the required
factorization u such that E o a; = x-
Let us now study the action of u; on the pseudo-elernents- Given c €*

1.10 Some diagram lemmas
47
X >-
-^ Z
^> C
J >—=—> Y
-^> W
Diagram 1.34
C, we have e(c) €* F and since 77 is an epimorphism, there is e £* E
such that 7?(e) =* e(c). Now (^ o A)(e) =* {fi o 77)(e) =* (n o e){c) -* 0,
so that we find g €* G such that t'(g') =* A(e). We shall prove that
n{g) =* uj{c).
By 1.9.5, the relation 7?(e) —* e(c) implies the existence oi z £* Z
such that A{z) =* c and rB;) =* e. Now
(H o uj){c) =* x(c) =* (x o A)(z) -* (A o A o r)(z)
-*(AoA)(e)=*(Aoi.)E)=*H7rE),
from which a;(c) =* 7r(fl') since S is a monomorphism.
We can now prove the exactness of the sequence (/3,a;,r) by chasing
on diagram 1.33. First of all choose b E* B. To prove that {ujof3){b) =* 0,
put c =* /3{b) and e =* 6{b) in the previous description of cj; one has
indeed 7y(e) =* (ry o S){b) =* (e o /3)F) =* £:(c). As before, we choose
g e* G such that u{g) =* A(e) =* (A o 6){b) =* 0. This implies ^ =* 0
since i/ is a monomorphism and thus a;(c) =* 7r(^) =* 0.
Now choose c e* C and construct e €* E, g e* G as indicated in the
description of cj in terms of pseudo-elements. Suppose 7r{g) =* 0. This
implies the existence of d €* -D such that 9{d) =* g. Since (A o C)(d) ==*
{uo0){d) =* u{g) =* A(e), we can consider an element written formally

48 Abelian categories
e - C{d) e* E such that A(e - C(d)) =* 0. Since (ry o Q{d) =* 0, one
has ri{e - C{d)) =* r?(e). Since A(e - C(d)) =* 0, there is b €* B such
that 6{b) =* e - C(d). Finally (e o /3)F) =* (r? o 6){b) =* 7?(e - C(d)) =*
7y(e) =* £(c) and since e is a monomorphism, /3F) =* c. This proves
already that (/3, cj) is exact.
Choose again c^* C and construct two pseudo-elements e, ^ as before,
(r o a;)(c) -* (r o 7r)E) =* {p o i/)(ff) =* {p o A)(e) =* 0.
Finally choose j € J such that r{j) =* 0. Since tt is an epimorphism,
we find g €* G such that 7r{g) =* j. From (p o i/)(^) =* (r o 7r)(^) =*
r[j) =* 0, we deduce the existence of e E* £" such that A(e) =* i^{g).
From (/i o 7y)(e) =* (^ o A)(e) =* (^ o i/)(^) =* 0, we find c €* C such
that e{c) =* 7y(e). Going back to the description of cj, we conclude that
j ^* nig) =* uj{c). U
Lemma 1.10.9 (The snake lemma) In an abelian category, consider
diagram 1.35 where the two central rows are exact and the squares A),
B) are commutative. There exist morphisms a, /3, r, (f making the
diagram commutative as well as a diagonal morphism uj such that the
sequence (a, /3, cj, r, (p) is exact.
Proof The existence of the exact sequences (a, /3) and (r, (f) is attested
by 1.10.1. Let us consider diagram 1.36 where C = C2 ^ Ci ^^^ ^ =
^2 0^1 are image factorizations. Applying 1.5.6, we find morphisms F, A
keeping the diagram commutative. We define A = KerF, E = KerA,
E = CokerF, T = Coker A. There are obvious factorizations ai^fSi^Ti, (fi
through the kernels and cokernels, with a = 0:2 o ai, /3 = /32 o /3i,
r = r2 o n and (p = (P20 (fi. 0L2 and ^2 are obviously monomorphisms,
and in the same way t\ , c^i are epimorphisms. Observe that by definition,
all the columns are exact.
Let us prove that ^2 is also an epimorphism, from which it will be an
isomorphism (see 1.5.1). Given c E* C, (^2 o A o e)[c) =* (/x o e){c) =* 0
so that (A o e){c) =* 0 since ^2 is a monomorphism. This implies the
existence oi u e* U such that E{u) =* e{c). Finally {e o f32){u) =*
E{u) =* e{c), and thus C2{u) =* c since e is a monomorphism. By
duality, ri is an isomorphism as well.
Applying proposition 1.8.2, we conclude that (C2?^) and (^', ^1) are
exact sequences, so that lenmia 1.10.8 can be applied to the central part
of the diagram, giving a morphism x such that the sequence (/3i, x? T2)
is exact. It remains to define a; = r{"^ 0x0 ^3^^ and, since ri and /32 are
isomorphisms, (/3, a;, r) is an exact sequence as well. D

1.11 Exact functors
0 0 0
49
■^ K
^
-> L
0
Diagram 1.35
> 0
1.11 Exact functors
Exact functors are those which preserve exact sequences. More precisely:
Definition 1.11.1 Consider an additive functor F: s/ >^ between
two abelian categories j/, J^. We say:
A) F is left exact when it preserves exact sequences of the form
0 >A >B >C;
B) F is right exact when it preserves exact sequences of the form
A >B >C >0;
C) F is exact when it preserves exact sequences of the form
0 >A >B >C >0.

50
Abelian categories
a2
-> T
T2
-> K
^1
-> S
^2
Diagram 1.36
-> L
Proposition 1.11.2 Consider an additive functor F: si >^ between
two abelian categories j/, ^. The following equivalences hold:
A) F is left exact iff F preserves finite limits;
B) F is right exact iff F preserves finite colimits;
C) F is exact iff F preserves finite limits and finite colimits.
Proof By duaUty, it suffices to prove A). If F preserves finite Umits, it
preserves kernels and thus is left exact (see 1.8.5). Conversely F preserves
biproducts since it is additive (see 1.3.4) and preserves kernels since it is
left exact (see 1.8.5). Applying 2.8.2, volume 1, it remains to prove that
F preserves equaUzers. But the equahzer of a pair /, g: A I B is just
the kernel oi f — g. D
Proposition 1.11.3 Consider an additive functor F: si >0i between
two abelian categories j/, ^. The following conditions are equivalent:
A) F is exact;
B) F preserves all exact sequences.
Proof B) => A) is obvious. Conversely, we consider an exact sequence
{f,g) and the image factorizations f = iop, g = joq asin diagram 1.37.
By 1.11.2, F preserves monomorphisms and epimorphisms. But the
sequence (i,g) is exact (see 1.8.2) and, by definition, is preserved by F.

1.12 Torsion theories
51
F{I) F{J
Diagram 1.38
> C
Diagram 1.37
F{A) ^ >F{B) ^ > F(C)
Thus we obtain diagram 1.38 in ^ with (F(z),F(g)) exact. Therefore
(F(/), F{g)) is exact (see 1.8.2). D
Proposition 1.11.4 Consider a left exact functor F: si >^ between
abelian categories. The following conditions are equivalent:
A) F is exact;
B) F preserves epimorphisms.
Proof A) => B) is already proved (see 1.11.2). So assume B) and
consider a short exact sequence
0
^A-
a
^B-
->C-
^0
in j/. Applying F we get an exact sequence
0
>F{A) ^^""^ >F{B) ^^^^ )F(C).
Since /3 is an epimorphism, so is F(/3) and the second sequence is in fact
a short exact sequence (see 1.8.5). D
1.12 Torsion theories
An element a of an abeUan group A is cyclic (or has a torsion) when there
exists a non-zero natural number n € N such that na = 0. An abeUan
group A is torsion free when it does not contain any non-zero cyclic
element; on the other hand A is a torsion group when all its elements
are cyclic.

52 Abelian categories
0 > A' > —> A ^—^> A''
F
Diagram 1.39
Given an abelian group A, the set A^ of its cycUc elements is a
subgroup of A. Indeed if na = 0 and mb = 0, for a,b ^ A and n^m E N*,
then nm{a + 6) = 0 and n{—a) = 0. Thus A' is a torsion group. On the
other hand the quotient A/A' is obviously torsion free, which yields an
exact sequence
0 >A' >A >A/A' >0.
Observe also that given a group homorphism /: A >B, if na = 0 in
A, then nf{a) = 0 in ^. Thus if A is a torsion group and B is torsion free,
/ must be the zero homomorphism, since 0 is the only cyclic element in
B.
This example is the leading one for understanding the definition of
a torsion theory. We recall that a full subcategory is replete when it is
closed imder isomorphisms (see 3.5.1, volume 1).
Definition 1.12.1 A torsion theory on an abelian category ^ is a pair
{T^T) of full replete subcategories of ^ such that:
A) every morphism f: T >F, with T € T and F € J^, is the zero
morphism;
B) for every B e ^, there exist T e T and F € T together with an
exact sequence
0 >T >B >F >0.
Observe that the previous definition is autodual.
Proposition 1.12.2 Consider an abelian category Si provided with a
torsion theory {T,J^). Under these conditions:
A) if f: A> >F is a monomorphism with F e J^, then A E J^;
B) ifg: T >C is an epimorphism with T eT, then C €T.
Proof By duality, it suflSices to prove the first statement. Consider an
exact sequence (fc,g) with A' G T, A" G ^ as in diagram 1.39. Since
A' G T, F G ^, fok = 0. Since q = Coker fc, one gets g such that goq = /.

L12 Torsion theories
r
St"
Diagram 1.40
53
■^ T ^zzzz^T'er"—^—> f
^ k ^
-> 0
Since / is a monomorphism, so is q. Since q is both a monomorphism
and an epimorphism, it is an isomorphism (see 1.5.1). Since A'^ E J^ and
^ is replete, AeJ^. D
Proposition 1.12.3 Consider an abelian category ^ provided with a
torsion theory (T^!F), Then:
A) T and T contain the zero object;
B) T and T are closed under biproducts.
Proof By 1.12.1 there exists an exact sequence
0 >T >0 >F-
^0
with T €T, F e J^. Therefore T >0 is a monomorphism (see 1.8.5);
but it is also an epimorphism since 0 is initial. Thus T >0 is an
isomorphism (see 1.4.1) and 0 E T since T is replete. By duaUty, 0 E ^.
Next consider T', T" E T. By 1.12.1 there exists a short exact sequence
(A:, g) with T €T, F € J^] see diagram 1.40. Considering the canonical
morphisms st', st" of the biproduct, one has q o st' = 0, q o st" = 0
since T,T' E T and F E ^. But k = Kerq (see 1.8.5), thus there exist
u,V such that kou = ST',kov = st". Since T' 0T" is a coproduct, we
get / such that lo st' = tx, / o st" = v. The relations
ko I o st' = kou = st', k ol o st" = k ov = st"
imply kol = It'^t"• Thus kolok = k and since A: is a monomorphism,
lok = It. So A: is an isomorphism and since T €T, one gets T'^T" € T.
The stability of J^ under biproducts follows by duality. D

54
-> T
Abelian categories
— > B > F
-> 0
^ T' TT—^ B -,—> F' > 0
k q
Diagram 1.41
Proposition 1.12.4 Consider an abehan category 0i provided with a
torsion theory (T, J^). For an object B e ^, the exact sequence
0 >T >B >F >0
as in 1.12.1 is unique up to isomorphism.
Proof Suppose we are given two such sequences {k^q) and {k'^q') as
in diagram 1.41. Since T e T and F' ^ J^^ q' o k = Q and we get a
factorization I such that k' ol = k^ because k' = Kerq\ Analogously we
get V such that koV = k'. Prom koVol = k'ol = k and k'oloV = koV = fc',
we deduce V o I = 1^ and I oV = 1 j'/ since fc, k' are monomorphisms.
So l^V are indeed isomorphisms making the diagram commutative. By
duality, one gets the corresponding result for F^F'. D
Proposition 1.12.5 Consider an abehan category ^ provided with a
torsion theory {T,T). Given an object B e ^,
A) BeT iff VF G ^ ^(B, F) = {0},
B) BeJ^ iff \fTeT ^{T,F) = {0}.
Proof By duality, it suffices to prove the first statement. If B G T,
^(B, F) = {0} for every F e T^ just by definition of a torsion theory.
Conversely suppose ^{B, F) = {0} for every F e J^ and consider the
following exact sequence, with T eT and F e T (see 1.12.1):
0-
^T-
^B-
^F-
^0.
By assumption, gf = 0 so that its image is just 0. By 1.8.5, q is already an
epimorphism, so that F is isomorphic to the image 0 of q. By 1.8.5 again,
k is both a monomorphism and an epimorphism, so it is an isomorphism
by 1.5.1. Since T eT with T replete, one concludes that B eT. D
In section 5.7, volume 1, we studied universal closure operations. We
shall now prove an interesting relation between universal cloBlire opersr
tions and torsion theories.

1.12 Torsion theories
a
0
■^ T
-> F
Diagram 1.42
55
E y-
0 >-
^ D ^
> Oc >
Diagram 1.43
-> B
^ C
Proposition 1.12.6 Consider an ahelian category 0i provided with a
universal closure operation. Define two full subcategories T^Tof^ by
A) TeT if 0 >T is dense,
B) FeT if 0 >F is closed.
Under these conditions, the pair (T, J^) is a torsion theory on ^.
Moreover this torsion theory satisfies the additional property:
If s: S> >T is a monomorphism and T eT^ then S eT.
Proof Obviously T and T are replete. Moreover ii T e T, F e J^
and /: T >F, consider diagram 1.42. Since a is dense and /3 is closed,
5.7.10 of volume 1 implies the existence of a unique 7 making the diagram
commutative. In other words, f = 0.
Now given an object B G ^, consider the subobject 0 >B, its
closure Ob > >B and the coequalizer B »C of Ob > >B. By 5.7.5
of volimae 1, 0>-
sequence
-^Ob is dense and thus Ob G T. We have an exact
0-
^Ob
k
■^B-
^C-
^0
with Ob G T. We shall prove that C ^T. Consider diagram 1.43 where
both squares are puUbacks and Oc is the closure of 0 in C. One has E =
g*~^@) thus E = Ker^. But Kerq = k by construction, so that E = 0^.
On the other hand D ^ E; see 5.7.1, volume 1. Thus D ^ 0^ ^ 0^ and

56 Abelian categories
Diagram 1.44
e is an isomorphism. Therefore p o e is an epimorphism and finally the
monomorphism 0> >0c is also an epimorphism, thus an isomorphism
(see 1.5.1). So 0> >C is isomorphic to Oc > >C, which is closed. Thus
Finally consider a monomorphism s: S> >T with T eT.
Considering the triangle of diagram 1.44, from the density of r we deduce that
of (t; see 5.7.7, volume 1. D
Definition 1.12.7 Consider an abelian category 3i provided with a
torsion theory (T, J^). This torsion theory is hereditary when it satisfies
the additional condition:
If s: S> >T is a monomorphism and T eT^ then S eT.
Theorem 1.12.8 Consider an abelian category ^. There exists a bijec-
tion between
A) the universal closure operations on 3i and
B) the hereditary torsion theories on 0i.
Proof In 1.12.6 we have constructed an hereditary torsion theory
from a given universal closure operation. Conversely, let us start with
an hereditary torsion theory (T, J^) of ^. Given a subobject A> >B^
we consider diagram 1.45 where the vertical and the horizontal sequence
are exact and T G T, F G J^; see 1.12.1. The pair (c, d) is the puUback
of (fc,6). Let us define the closure of a: A> >B as being d: A> >B.
Since 6oa = 0 = fcoO, there exists e such that doe = a, coe = 0. This
proves already that AC A.
Choose now AC A^ C B. Write u: A> >A^ for the inclusion; we shall
prove that A C A', For this consider diagram 1.46 where fc, ^, a, 6, c, d are
defined as in diagram 1.45 and k'^q'^a'^b'^c'^d' are defined analogously
from a'. Since 6' o a = 6' o a' o n = 0, we get a factorization v through
6 = Cokera, yielding woe = d ox. Since T gT and F' G J^, q' ovok = 0

1.12 Torsion theories
0
57
-> T > —>B/A —» F
-> 0
0
Diagram 1.45
from which we get the factorization w through k^ = Ker^', yielding
V o k = k' o w. Finally k'owoc = vokoc = vohod = h' o d^ from
which we get the factorization x through the pullback {c'^d')^ yielding
w o c = c' o x^ d = d' o X. The existence of x indicates precisely that
Consider again A C B as in diagram 1.45. We shall prove that A = A.
For that consider diagram 1.47 where z = Cokerd. From the equality
q o b o d = q o k o c = 0, we obtain a factorization p with p o z =
qob. The arrow p is an epimorphism since q and b are. But the square
fcoc = 6odisa pullback, by definition of A; by lemma 1.10.2, p is then
a monomorphism, thus an isomorphism; see 1.5.1. Therefore the kernel
of p is q>—^B/l; see 1.1.7. Now 0 € T by 1.12.3, so that diagram 1.48
defines A via its pullback square. But A = z~^{0) is precisely the kernel
of z, i.e. d: 1 >B, Thus A = A
Finally let us consider a subobject a: A> >B and an arbitrary mor-

58
Diagram 1.46
^ 0
A >—-—> B ^-^>B/A
T > —>B/A —» F
Diagram 1.47

1.12 Torsion theories
0
59
-> B
-> 0 > >B/A>—-—» F
-> 0
Diagram 1.48
phism /: C >B. We consider diagram 1.49 where {a^^g) is the pull-
back of (/, a), a, 6, c, d, fc, g' are defined as in diagram 1.45 and a', b\
c', d', fc', gf' are defined analogously from the subobject f~^{A)> >C.
Prom the relation bo foa^ = boaog = Owe get the factorization h such
that bo f = hob' ^ because b' = Cokera'. Since the square f oa' = ao g
is a puUback, /i is a monomorphism (see lemma 1.10.2). Since T' e T
and FEJ^^qohok' = 0 from which we get the factorization t through
k = Kergf, yielding hok' = kot. The arrow t is a monomorphism since h
and k' are. Prom bofod! = hob'od! = hok'oc' we obtain the factorization
5 through the puUback (c,d) of (fc,6), yielding f od! = do5, toe' = cos.
This completes the diagram. Let us now consider diagram 1.50, where
the square is a puUback. Since h o k' = k o t^ we get a factorization I
making the diagram commutative, i.e. k' = no I and t = mo I. Since
T e T and the torsion theory is hereditary, S e T. But since S G T
and F' e ^, ^' o n = 0. So we get a factorization r through k' = Kerq'^

60
Abelian categories
f-'{A)>
d
d!
_ d
A >
'", ifc'
T'>-
b'
■^C/f-HA)
^ B
b
-^>F'
T> > B/A ^
Diagram 1.49
-*F
T > , >B/A 5-^ F
k '"'" Q
Diagram 1.50

1.12 Torsion theories 61
A' > > A > > B
0 > > 0 > >B/A »{B/A)/0 > 0
Diagram 1.51
yielding k'or = n. Therefore k'orol = nol = k'^ from which ro/ = 1 j',
since k' is a monomorphism. In the same way nolor = k'or = n and
lor = ls since n is a monomorphism. So Z,r are isomorphisms and the
square kot = hok' is d. puUback. But the squares k' o c' = h' o d! and
k o c = b o d are also puUbacks by definition of the closure operation.
Applying 2.5.9 of volume 1, (associativity of puUbacks), we deduce that
dos = fod^isa. puUback, or in other words f~^{A) = f~^{A).
It remains to prove that we have defined a bijective correspondence.
Let us start with the universal closure operation and consider the pair
(T, J^) defined in 1.12.6. Write (—) for the original closure operation and
(—) for that deduced from the pair (T, J^), as in diagram 1.22. Given a
subobject A> >B^ consider diagram 1.51 where the bottom sequence
is exact with 0 G T and {B/A)/0 e T\ see proof of 1.12.5. The pair
(c, d) is the puUback defining A and the left-hand square is defined as
a puUback, i.e. e = Kerc. Since 6~^@) = Kerb = A, one has in fact
Al — A. By universality of the original closure operation (see 5.7.1.D),
volume 1), one obtains A = A^ = A. ^
Conversely start with an hereditary torsion theory (T^J^)^ construct
the associated closure operation (—) and the corresponding pair (T', J^'),
as in 1.12.6. Given B e ^ consider the closure 0> >B of 0> >B as
constructed in diagram 1.45. Since the cokernel of 0> >B is just the
identity on B (see 1.1.8), this closure is given by the puUback square
of diagram 1.52 where the bottom sequence is exact, with T e T and
F E !F; see 1.12.1. Since the puUback of an identity is an identity, c= It
and thus 0 = T. Since 0 is dense in 0, 0 G T'; see 1.12.6.
li B eT, then 0 = B; see 1.12.6. Thus B = 0 = T eT. Conversely
if B E T, the sequence
0 >B ^—>B >0 >0
is exact with B eT and 0 E ^; see 1.12.3 and 1.8.5. Thus J3 = T by

62
0 >-
Abelian categories
d
-> B
-> T >-
-> B
-» F
-> 0
Diagram 1.52
uniqueness of such a sequence (see 1.12.4). But then B = T = 0 eT\
Since in a torsion theory (T, J^), the class !F is completely determined
by T (see 1.12.5), T = T implies J^ = J^\ D
It should be observed that in the construction of the universal closure
operation associated with an hereditary torsion theory, the hereditary
condition has been used just to prove the axiom f~^{A) = f~^[A). So
an arbitrary torsion theory induces a closure operation satisfying the
three axioms:
A) 5 C 5;
B) 5 C T
C) S = S.
SCT;
But it is false in general that such a closure operation is induced by a
torsion theory (see exercise 1.15.15).
1.13 Localizations of abelian categories
For "good" abelian categories ^, the localizations of ^ (see 3.5.5, vol-
lune 1) are exactly described by the hereditary torsion theories on ^ or,
equivalently, by the universal closure operations on ^ (see 1.12.8). In
this section, "good" will mean "locally finitely presentable". The reader
should refer to chapter 5 for the theory of locally presentable categories.
In fact this assumption will just be used in this section through the
references to chapter 5, volume 1.
In an abelian category, the bidense morphisms (see 5.8.1, volume 1)
with respect to a universal closure operation can be described in a
somehow simpler way.
Proposition 1.13.1 Consider an abelian category ^ provided with a
universal closure operation. Given a morphism f of J^, the following
conditions are equivalent:

1.13 Localizations of abelian categories 63
0 > A -^^-^ P ^^^^ A -^-^ I > 0
y
Diagram 1.53
A) f is bidense;
B) (a) the image of f is dense,
(b) the monomorphism 0 >Ker/ is dense.
Proof Consider the image factorization f = top oi f. In both cases i
is required to be dense.
Since i is a monomorphism, the kernel pair (ix, v) of / coincides with
that of p. But since p is an epimorphism and every epimorphism is
regular (see 1.4.1), p = Coker (ix, v) (see 2.5.7, volume 1) or in other words
p = Coker (ix — v); see 1.2.8. On the other hand if fc = Ker(ix, t;), k =
Ker (u—v); see 1.2.8. Therefore the upper exact sequence in diagram 1.53
is exact.
Considering the image J oiu — v^we know that j = Kerp; (see 1.8.1)
and q = Coker fc; see 1.8.3. Since i is a monomorphism, the kernel of / is
also that of p, thus it is j. Since q = Coker fc and fc is a monomorphism,
k = Kerq; see 1.5.7. Thus k = ^~^@) and the square is a puUback. If y
is dense, so is k (see 5.7.4, volume 1) and if k is dense so is y; see 5.7.8,
volume 1. n
Proposition 1.13.2 Consider an abelian category 3i provided with a
universal closure operation. Write (T, !F) for the corresponding
hereditary torsion theory (see 1.12.6). Given a morphism / G ^, the following
conditions are equivalent:
A) f is bidense;
B) the objects Ker / and Coker / belong to T.
Proof For a morphism /: A >B, consider the situation of
diagram 1.54, where k = Ker/, q = Coker/ and / = i op is the image
factorization. Observe that / »0 is an epimorphism since 0 is initial
and dually, 0 >Q is a monomorphism since 0 is final. We know that
p = Cokerfc and dually i = Kerq; see 1.5.5. But ^"^@) = Ker^ = z, so
that the square is a puUback.

64 Abelian categories
0 > K >—-—> A —> B ^> Q > 0
/
^> 0
Diagram 1.54
If i is dense, so is j because g^ is a (regular) epimorphism (see 5.7.8,
volume 1) and conversely if j is dense, so is i; see 5.7.4, volume 1. But /
is bidense iff 0 >K and i are dense (see 1.13.1) thus iff 0 >K and
0 >Q are dense, i.e. iff K and Q are in T; see 1.12.6. D
Definition 1.13.3 Consider an abelian category ^ and a full replete
subcategory T C ^. Define a class E C ^ of morphisms by
/gS iff Ker/GT, Coker/eT.
The subcategory T is called ^%calizing^^ when E is the class of those
morphisms of 0i inverted by a localization r -\i: si^ y^.
Let us prove now that for a locally finitely presentable abelian category
^, if (T, J^) is an hereditary torsion theory, T is a localizing subcategory.
In order to apply theorem 5.8.8, volume 1, we first prove the following
result.
Proposition 1.13.4 If ^ is an abelian category provided with a
universal closure operation, the class of bidense morphisms is stable under
hnite colimits (see 5.4.9, volume 1).
Proof We consider a finite category S>, two functors F, G: S> >^ and
a natural transformation a: F =^ G such that, for each object D e S^,
a£>: F{D) >G{D) is bidense. We must prove that the factorization
colim a: colim F > colim G
is bidense.
A finite colimit can be computed from finite coproducts and coequal-
izers (see 2.8.1, volume 1), thus from finite coproducts and cokernels (see
1.2.8).
The case where ® is empty is obvious, since both colimits are the zero
object and the factorization is just the identity on it.
The case of a finite coproduct follows at once from the case of a
binary coproduct, just by associativity of coproducts (see 2.2.3, vol-

0
1.13 Localizations of abelian categories
> Ao
65
Diagram 1.55
fci © ^2 ai © a2
-^Ki © K2 >Ai © A2 >Bi © B2
Pl©P2
31^32
Diagram 1.56
ume 1). Coproducts are biproducts (see 1.6.5). Therefore given two
bidense morphisms ai,a2 as in diagram 1.55, their images a^ = ji op^
and their kernels fci, let us compute the various biproducts to get
diagram 1.56. By interchange of products and kernels (see 2.12.1, volume 1),
ki®k2 = Ker (ai © a2) and by interchange of coproducts and cokernels,
Pi®P2 = Coker (fci © /u2). In particular ji © J2 is the image of ai © a2.
To conclude the proof in this case, it suffices to prove that the mono-
morphisms ji © J2 and 0 © 0 are dense.
It is indeed a general fact that the biproduct of two dense monomor-
phisms ji, J2 is again dense. It suffices to consider diagram 1.57 where
both quadrilaterals are puUbacks. Since the bottom monomorphisms are
dense, the top ones are dense as well (see 5.7.4, volume 1) and thus also
their composite (see 5.7.7, volume 1).
It remains to prove that bidense morphisms are stable under cokernels.
For this consider diagram 1.58 where 9oa = /3of^q = Coker/ and
p = Coker^. Prompo/?o/ = pogoa = 0 we find 7 such that joq = po/3.
Assuming a, C bidense, we must prove that 7 is too.
First of all, we reduce the problem to the case where /, g are
monomorphisms. For this we factor / and g through their images / = /2 o /i,
9 = 92^ 9i and we consider the corresponding factorization 6: I >J
given by 4.4.5, volume 1, and making the left-hand squares of disr

66
Abelian categories
h e /2> >h ® ^2> >Bi e B2
A =^—> B ^—^> Q
a
P
-> D
9 ^ P
Diagram 1.58
-^> P
gram 1.59 commutative. Since /i and gi are epimorphisms, we still have
q = Coker/2, p = Coker5'2 with, now, /2, g2 monomorphisms and C
bidense; we want to prove that 6 is bidense as well. Considering the
image factorizations a = a2 o ai^ C = C2 o /3i, 6 = 62 o 61 of a, /?, 6, we find
again by 4.4.5, volume 1, factorizations /ii, /i2 making the corresponding
squares of diagram 1.60 commutative. Since gi is an epimorphism and
a2 is dense, 62 is dense by 5.7.8, volume 1. Next we compute the
kernels ao, /So, ^0 of Q;? /?, ^5 which are also those of ai, /?i, 71 since a2, /?25 72
are monomorphisms. Obvious diagram chasing (like in 1.10.1) shows the
existence of morphisms ki, ^2 making the corresponding squares of
diagram 1.60 commutative. 0 ^Iq is a monomorphism because 0 is final
while ^2 is a monomorphism because so are 60 and /2. 0 >Bo is dense
because /3 is bidense (see 5.8.1, volume 1), thus 0 >Io is dense as well
(see 5.7.7, volume 1). This concludes the proof that 6 is bidense.
Next we split the problem into two parts: the case where 6, C are both
epimorphisms and the case where they are both monomorphisms. For
this we refer further to diagram 1.60 and consider r = Coker/12. Notice
that /i2 is still a monomorphism since so are 5^25 ^2- Obvious diagram
chasing produces 71,72 making the diagram commmutative and from
72 071 o^ = 72oro/3i =po j32 0 Ci = po P = joq
we deduce 7 =: 72 o 71 since g is an epimorphism. By 5.8.4, volume 1,

1.13 Localizations of abelian categories
A —=^-^> I >—-—> B —y> Q
67
a
/3
-* J > TiZ > D
9i " 92 ^ P
Diagram 1.59
^> P
0
0
0
h
-y lo
k2
-> Bo
-> Qo
do
00
70
A » I ) > B » Q
ai
X
6i
-^ y V-
/3i
h2
-> Z
71
-» R
Ol2
C
02
91
-» J >
92
Diagram 1.60
» D
h'2
-» P

68
Abelian categories
J >-
-> D
92 ^ P
Diagram 1.61
-> 0
» A^
n
» R
72
-^> P
it suffices now to prove separately that 71 and 72 are bidense. As far
as 72 is concerned, observe that r = Coker/12 and p = Coker^2 with
^2,5^2 monomorphisms, while 62,/?2 are bidense monomorphisms, since
6, /3 are bidense (see 5.8.1 and 5.8.2, volume 1). As far as 71 is concerned,
observe that q = Coker/2 and r = Coker/12 with /2, /12 monomorphisms,
while 61,/?i are bidense epimorphisms since 6^/3 and 62,/?2 are bidense
(see 5.8.4, volume 1).
We treat first the case of 71. Since r and /3i are epimorphisms, so is
7i. Thus the image of 71 is just the identity on i?, which is dense by
5.8.4, volimie 1. We complete diagram 1.60 with 70 = Ker7i and the
corresponding factorization s: Bq >Qo such that Jqos = qo/3o. Since
^2?/?2 are monomorphisms, one still has 60 = KerSi and /?o = Ker/?i.
By 1.5.7 and 1.8.5, we thus have short exact sequences F05^1M (/^Oj/^i)?
G0,71), (/2,9), {h2^r). Applying the "nine lemma" (see 1.10.5), we
conclude that (/u2, s) is a short exact sequence as well. In particular, s is an
epimorphism and since 0 >B is dense by bidensity of /?, 0 >Qo is
dense by 5.7.8, volume 1. This concludes the proof that 71 is bidense.
To prove that 72 is bidense, we observe first that since C2 is a dense
monomorphism and p is an epimorphism, the image of 72 is dense by
5.7.8, volume 1. Next we construct diagram 1.61 where n = Ker72 and

1.13 Localizations of abelian categories 69
(m, t) is the puUback of (r, n). The arrow t is an epimorphism since r is
(see 1.7.6). Prom
po/?2o?7i = 72oro?7i = 72ono^ = 0
we get a unique u: M > J such that g2 o u = C2 o m^ because g2 =
Ker (Coker^2) = Kerp; see 1.5.7. The pair (Z, v) is defined as the puUback
of F2, ix), so that I is a dense monomorphism because so is 62] see 5.7.4.
Observe next that
/32omol = g2 0uol = g2o62 0v = /32oh2 0v
from which mol = h2 0v^ since /?2 is a monomorphism. Therefore
notol = romol = roh2ov = 0
since r = Coker/12. This implies ^ o / = 0 because n is a
monomorphism. So the upper square in diagram 1.61 is commutative, with I a
dense monomorphism and t an epimorphism. Again by 5.7.8, volume 1,
0 >N is dense and finally, 72 is bidense. D
Theorem 1.13.5 Consider a locally finitely presentable abelian
category ^. There are bijections between:
A) the localizations of ^;
B) the universal closure operations on ^;
C) the hereditary torsion theories on ^;
D) the localizing subcategories of ^.
Proof By 5.8.8, volume 1, and 1.13.4, this volume, we have a bijection
between the localizations and the universal closure operations. By 1.12.8,
there exists also a bijection between the universal closure operations and
the hereditary torsion theories.
Every localization corresponds to an hereditary torsion theory (T, J^)
and the morphisms / inverted by the reflection are precisely those with
Ker/ e T, Coker/ e T; see 1.13.2, this volume, and 5.8.8, volume 1.
Thus T is a localizing subcategory. Observe that two different
localizations induce different localizing subcategories, just because each one of
the classes T, J^ characterizes the other one (see 1.12.5). We have thus
defined an injection from the class of localizations to that of localizing
subcategories.
But this is also a surjection. Indeed every localizing subcategory T, by
definition, induces a localization. Considering the corresponding xmiver-
sal closure operation, a monomorphism 0 >B is inverted when it is
bidense (see 5.8.8, volume 1), thus when it is dense (see 5.8.2, volume 1).

70 Abelian categories
On the other hand this monomorphism is inverted iff its kernel and its
cokernel are in T; see 1.13.3. The kernel of the monomorphism 0 ^B
is 0 0 (see 1.1.7) and the cokernel of the zero morphism 0 >B
is B B (see 1.1.8). Since 0 0 is inverted by the localization, its
kernel and its cokernel are in T, thus 0 G T. So finally 0 >B is dense
iS B eT. Applying 1.12.6, we conclude that the hereditary torsion
theory associated with the localization has indeed the form (T, J^), for T
the original localizing category. D
Counterexample 1.13.6
An abelian group T is divisible when for each a E T and each n G N*,
there exists b eT such that nb = a. An abelian group F is reduced when
the only divisible subgroup of F is @). Write T for the full subcategory
of divisible groups and T for the full subcategory of reduced groups. We
show (T, J^) is a torsion theory, but not an hereditary torsion theory.
First of all observe that given a divisible abelian group T and a
morphism of abelian groups /: T >F, the image /(T) is still a divisible
group. When F is reduced, /(T) = 0, proving that / = 0.
Next observe that given a family of divisible subgroups T^ C A, z G /,
their sum H"ie/ Ti C A is still a divisible subgroup. Indeed an element
t G "hie/^i has the form ti-\-.. .-\-tj with tk G T^^; given n G N*, choose
Sk G Tij^ such that nsk = tk- It follows that n{si -\-...-\- Sj) = t. Given
an abelian group A, write T for the biggest divisible subgroup of A, i.e.
the sum of all the divisible subgroups of A. Consider the exact sequence
0 >T> ^—>A 2—^yA/T >0.
Let us prove that the quotient A/T is reduced. If X C A/T is a divisible
subgroup, then q~^{X) is divisible since, given a G q~^{X) and n G N*,
we find [a] G X and thus [b] G X with n[b] = [a]. This means that
b G q~^{X) with nb - a e T. Since T is divisible, choose t G T such
that nb — a = nt. This indicates that a = nb — nt = n{b — t). But
q{b-t) = q{b) - q{t) = q{b) G X, thus b-te q~'^{X). This proves that
g~^(X) is divisible, and thus contained in T. As a consequence, X = @)
and A/T is reduced.
Thus (T, T) is a torsion theory. It is not hereditary because the group
Q of rational numbers is divisible, while the subgroup Z of integers is
not.

1.14 The embedding theorem 71
1.14 The embedding theorem
This section is devoted to the proof that every small abelian category
has an exact, full and faithful embedding in a category of modules. Let
us first observe that:
Proposition 1.14.1 Given a small abelian category and a filtered
diagram of left exact additive functors Ff. si >Ab, the pointwise colimit
colim Fi is again left exact and additive.
Proof By 2.12.1, volume 1, an arbitrary colimit of functors preserving
finite coproducts is again a functor preserving finite coproducts.
Therefore colim Fi is additive (see 1.3.4).
By 2.13.6, volume 1, a filtered colimit of functors F^: si >Ab
preserving finite limits is again a functor preserving finite limits. Thus
colim Fi is left exact (see 1.11.2). D
Given a small abelian category si ^ we construct first a faithful and
exact embedding G: si >Ab in the category of abelian groups. By
1.3.8, IJ caii in any case be described as a colimit JJ = colim j/(i4i, —)
of additive Ab-valued representable functors. We shall thus define U as
the colimit of a composite
V ^—>si ^—>Add(j/,Ab)
where
A) y is the contravariant Yoneda embedding defined on the objects by
Y{A)^^{A,-),
B) V and 0 are chosen in a way which ensures the exactness and
faithfulness of U.
Lemma 1.14.2 With the previous notation, U is left exact as long as
V is co&ltered.
Proof Each representable functor si{A^—): si >Ab is left exact,
because the representable functor si{A^ —): si >Set is (see 2.9.5,
volume 1) and the forgetful functor Ab >Set reflects limits (see 2.9.8.d,
volume 1).
Since Y is contravariant, the cofilteredness of V implies that C/ is a
filtered colimit of representable, thus left exact, functors. So U is left
exact by 1.14.1. D

72
Abelian categories
/3
-> B
Diagram 1.62
a
H{D)
^> C
Diagram 1.63
Lemma 1.14.3 With the previous notation and supposing that the
category V is cofiltered, U preserves epimorphisms as soon as every
epimorphism of the form f: A >(t>{D) in si can he written as f =
(f>{d), with d: D' >D in V.
Proof Consider an epimorphism g: B >C in si\ U{g) is an
epimorphism precisely if it is a surjection (see 1.8.5.e, volume 1). Recall that
filtered colimits in Ab are computed as in Set (see 2.13.6 and 2.13.3,
volimie 1).
Choose now x G U{C) = colimj/@(J9),C). The element x is thus
the class of some arrow 7: 0(J5) >C For U{g) being surjective, we
must find an element y e U{B) = colimsi[(j){D),B) mapped to x. The
element y is the equivalence class of some arrow /3: (t>{D') >B and
the relation U{g){y) = x means that g o /3 and 7 are equal "at some
further level J9'"'. More precisely, U{g){y) = x ii one can find J9'' G V
and d: D" >D, d': D" >D' such that gopo(j){d') = 700(d), as in
dia^am 1.62.
Let us consider the pullback of diagram 1.63. By 1.7.6 / is an
epimorphism and, by assumption, can be written as / = 0(d), with d: D >D
in V, Putting D' = D, )9 = a, D'' = D, d' = 1;^ and d = d, the required
conditions for having U{g){y) = x are now satisfied. D
Lenama 1.14.4 With the previous notation and supposing V coGltered,

1,14 The embedding theorem 73
U is faithful as long as
A) every object A e s/ has the form A = (t>{D) for some D eV,
B) for every morphism d ^V, (j){d) is an epimorphism in si.
Proof Consider two morphisms f,g: A I B in s/ such that U{f) =
U{g). Since U{A) = colimsi{(j){D), A), an element a e U{A) is the
equivalence class of some arrow a: (t>{D) >A. The equality U{f){a) =
U{g){a) means that / o a and g o a are equal "at some further level
J9"'. More precisely, U{f){a) = U{g){a) means the existence of an arrow
d: D' > J9 in V such that / o a o 0(d) = goao (t){d).
Now assume u{f) = U{g). If A is of the form (t>{D) for D e V,
1a'' <t>{D) >A corresponds to some element a G U{A). By the first
part of the proof we can find d G P such that /ol^o0(d) = 5'ol^o0(d),
which implies f = g \i (j){d) is an epimorphism. D
Theorem 1.14.5 (The faithful embedding theorem)
Every small abelian category admits a faithful and exact embedding in
the category of abelian groups.
Proof Using the previous notation, it remains to construct a small
category V and a functor 0: V >s/ which satisfies the conditions of
the three previous lemmas. We shall do this by constructing a sequence
(indexed by the natural numbers) of posets
and a corresponding sequence of functors 0^: Vn >s/, satisfying the
following conditions:
A) each Vn is a A-semi-lattice;
B) if n < m, 0n and 0m coincide on P^;
C) Vn G N "id eVn (t>n{d) is an epimorphism;
D) VA G j/ 3D G Pi (t>i{D) = A;
E) ^Aes/ ^DeVn V/: A »(l>n{D)
f epimorphism =>3d e Pn+i, d: D' >J9, 0^+i(d) = /.
Defining V = UneN^w ^^^ S^^^ ^^ ^^^^ ^^ extension 0: V >si of
the various 0^ and it is then obvious that the pair (P, 0) fulfils the
requirements.
We define Vq to be the one-point poset {*}; (f>o{*) is just the zero
object of s/. The poset Vq is obviously an A-semi-lattice and, since its
only arrow is 1*, its image under (f> is clearly an epimorphism.

74 Abelian categories
Suppose Do? • • • ,2^n have been defined and satisfy conditions A), B),
C), E). Consider all the pairs (i5,/) where D e Vn and f e s/ is an
epimorphism with codomain (j)n{D), We index those pairs by the
successive successor ordinals. We shall now construct a sequence of posets,
indexed by the ordinals, up to the supremum of the ordinals used for
the indexation,
and a corresponding sequence of functors 0": P" >j/, satisfying the
following conditions:
(a) each P" is a A-semi-lattice;
(b) if /? < a, 0^ and 0" coincide on P^;
(c) Va Vd G P", 0"(d) is an epimorphism;
(d) if the ordinal a indexes the pair (J5,/) with /: A >(l)n{D)^ there
exists d: D' >D in P^ such that 0^(d) = /.
Defining Pn+i = Ua^n ? we get an obvious extension dn+i- Pn+i >«^
of the various 0" and it is then clear that the pair (P^+i, 0nH-i) satisfies
conditions A), B), C), E).
We define (P^,(?!>^) to be {Vn,(t>n)\ {T^n^Vj satisfies conditions (a),
(b), (c) because {Vn^(l>n) satisfies conditions A), C). For a limit ordinal
a, P" = |J/3<a^n and 0" is the obvious extension of the various 0^. It
is immediate that ^" satisfies conditions (a), (b), (c) as soon as they
are satisfied for every /3 < a. Now suppose (P", 0") is defined and a -h 1
indexes the pair {D,f). We consider
iD = {D' eV^\D' <D}
and perform the disjoint union P" II [D, For simplicity, when D' < D
in P^, we write J9'* for the corresponding copy of D' in the second
component [D of the disjoint union. On P" II |J9, we provide P^ and
jjD with their original ordering and moreover we impose the relations
£)/* ^ jT)/ £qj. g^jj jT)' < J9^ The poset structure generated in this way
is given by the relations vaUd in P^ and [D together with the relation
D'* < D" for all elements D' <D,D' < D" in P^. We choose P^ U jD
with that poset structure as P""^^. Since P" is a A-semi-lattice, |J9 is a
A-semi-lattice as well. Now choosing Di e ID and D2 eV^, we compute
the infimum Di A JD2 in P^ and {Di A JD2)* = jDJ A JD2 in P^+^
Let us define 0^"*"^, which coincides with 0" on P^. We put
<+i(D*) = ^ and <-^ip* <£>) = /.

1.14 The embedding theorem
75
4>t'\D'y
X
K{D')
KiD')-
^> A
K{D' < D)
Diagram 1.64
-^> cj>l+\D"*)
Hn{D)
■^>KiD")
■^> A
4>1{D'<D") -^"^ ' K{D"<D)
Diagram 1.65
/
■^> MD)
liD' <D in P^, we define ^^+^{D'*) by the pullback of diagram 1.64,
which takes place in s/. Notice that ^n(-D) = <l>n{^)i t^^^s
considering </>"(!?' < D) makes sense; since this morphism and / are in
fact epimorphisms, u and v are epimorphisms as well (see 1.7.6). Since
A = <t>l{D% we put u = <+n^'* < ^*)- Since <(D0 = C^H^O,
we put ^ = 0^+1 (D'* <D'),
If D^ < D" < D in P", let us consider diagram 1.65 where both
squares are puUbacks and thus all morphisms are epimorphisms. Since
the outer square is a pullback and the lower composite is (t>^{D' < J9),
one gets P = 0^+i(D'*) and x = C+H^'* < ^0- We put y =
Finally ii D' < D and D' < D" in P^, let us define
C^H^'* < D") = <+n^' < D") o C+\D'* < D').
Just by associativity of puUbacks, 0""*"^ is a functor. By definition, each
0^"^^(d) is an epimorphism and 0""^^ extends 0", thus every 0^ with
0 < a. And also by definition we have the relation J9* < J9 in P""^^
with 0^+1 (jD* <D) = f, which takes care of condition (d).
It remains to verify condition D). The construction of Pi is indexed by
aU the pairs (*, /), where /: A >0o(*) = 0 is an epimorphism. Since 0
is initial, every morphism to 0 is an epimorphism and the construction is
finally indexed by all the pairs (*, /), where /: A >0 is any morphism.

76 Abelian categories
Since 0 is terminal, there is exactly one such / for each object A e s/,
thus condition D) follows from condition (d) in the construction of the
poset Pi. n
Let us observe that the diagram (P, 0) constructed in 1.14.5 has some
nice additional properties.
Lemma 1.14.6 With the notation of theorem 1.14.5, given two elements
J9i, J92 ^ P ^ie segment
[DuD2] = {DeV\Di<D< D2}
is Rnite.
Proof We constructed P by an induction Pq C P^ C ... on the natural
numbers and each Pn+i has itself been constructed by a transfinite
induction P" C P""*"^ C Let us prove that for two elements i5i,-D2
appearing at some level of the construction
A) [Di,D2] is finite,
B) the segment [J9i, J92] will remain unchanged at all further levels of
the construction.
Condition A) is certainly satisfied by Pq = {*}. Consider Di^D2
in P^"*"^ and suppose P" satisfies condition A). Let us observe that
the construction of P""*"^ implies that one has never J9' < J9*, for a new
element D* and an element D' eV^. Therefore if Di, D2 G P^, [Di,D2]
in P^"*"^ is just [J9i, J92] in P"; thus this segment is finite by induction
and condition B) has been verified for the passage from P" to P""^^. If
Di, D2 ^ P^, then Di = D* and D2 = D'* for D, D' G P^; [i?i, 1^2] in
P^+^ is isomorphic to [D, D'] in P^ and thus is finite. If Di ^ P^ and
jD2 G P^, then Di = D* and [Di,D2] = [D\D^] U [D, D2] where both
subsegments are finite. The case Di < D2 with Di G P" and D2 ^ P"
is impossible as we have seen.
Thus conditions A), B) remain satisfied at each successor step.
Because of that, they remain obviously satisfied at a limit step. D
Lemma 1.14.7 With the notation of 1.14.5, given elements Di < D2
in V, the canonical factorization
(t){Di) >limDi<D<D2 0(^)
is an epimorphism.
Proof Again we prove this by induction on the level at which the object
jDi has been introduced. Since at a given level of the induction one never
introduces elements greater than already existing elements, when Di is

1.14 The embedding theorem 77
A)
B)
C)
Diagram 1.66
> (t>{Doo)
introduced, D2 will be present as well and thus also all the J9's such that
Di<D< D2.
li Di e Vo^ Di = * = D2 and the hmit is the empty one. The
factorization indicated is thus the identity on the zero morphism, which
is indeed an epimorphism.
An object can never be introduced at a limit level in the construction.
So it remains to consider the case where Di is introduced at the level
P^+^ There is some Dq G V^ such that ^1 = ^5. There are two
possibilities for D2: namely, D2 G P" or D2 ^V^.
If Di = D2 ^ P", the limit is that of the empty diagram and the
factorization is the canonical morphism (t>{Di) >0, which is an
epimorphism since 0 is initial.
If Di ^ D2 ^ V^, D2 = DIq for some Dqq e V^. Consider
diagram 1.66. By construction of (P, 0) each square C) is a puUback and,
by definition of a limit, (j){D < Doo)opD = PDqq and 0(i5* < J5oo)opD* =
Pd*q' So there is one single composite rectangle B)-C), independent of
the choice of D. This rectangle is a puUback since it is just the limit of
the puUback rectangles C) (see 2.12.1, volume 2). But since the outer
rectangle is also a puUback by definition of (P, 0), we conclude by 2.5.9,
volume 1, that rectangle A) is a puUback as well. By the induction
hypothesis, / is an epimorphism and thus /* is an epimorphism as well
(see 1.7.6).
There remains the case Di = Dq, D2 eV^. Suppose the index a + 1
corresponds to the pair {D,J) (in the notation of 1.14.5). If Dq = D,
{D\D,<D<D2}=lD,D2]
has an initial object Z), so the corresponding limit is just (l>{D) (see
2.11.4, volume 1) and the factorization of the statement is just the
epimorphism /.
Finally let us suppose jDi = jDq < D* and D2 € P". We shall prove

78
Abelian categories
4>{Do)-
^<t>{Dl)
^<j>{D3)
Diagram 1.67
)■
that in this case the limit indicated is just 0(J5i) and the required
factorization is the identity. Put Ds = D2 AD^ which is an element of P".
li Di < D < D2 then, since Di <'D, Di < D a'D < D2 a'D. This
proves that
{D\ Di<D< Ds} is initial in {D\ Di < D < D2}
so that it suffices to compute the Umit on this initial part (see 2.11.2,
volimie 1). We want thus to prove that the cone
{cI>{Di),cI>{Di<D)d,<d<Ds)
is a limit cone. To do this, take another cone
{A,{fo:A ^<^P))p,<o<,,3
In diagram 1.67, the square is a puUback by definition. Since Dq < D^^
one also has Dq < Dq < Ds and Dq < D^ < Ds by construction of the
poset V. Thus the outer part of diagram 1.67 commutes, from which we
get a unique factorization /: A >(t){D*) making the whole diagram
commutative. Let us prove that this (unique) factorization works for
every index D of the given cone. If jDq < D < Ds, one can have D eV^
or D^V^.lfDe V^, then Do < jD and one has
(t>{Dl <D)of = cf>{Do < D)o(t>{Dl < Do)of = 0(Do < D)ofi,^ = /^
since Dq and D are indices in the original cone. U D ^ P", then
D = D'* with D' e V^ and D^ < Dq < D' < D3. We can then
consider diagram 1.68 where both squares are pullbacks by definition.
Since Do, D^ D'*,D3, DJ are indices in the original cone,

1.14 The embedding theorem
79
<l>iDo)-
-^4>{D')-
4>{Dl)
■^HDs)
Diagram 1.68
cl>{D'* < Dl) o ct>{Dl < D'*) o / = 4>{D'* <Dl)of
^Siy,=<t>{D'*<Dl)ofo>,
(j>{D'* < D') o <t>{Dl < D'*) o / = </,(J9o < D') o <^(D* <Do)of
= <l>{Do < D') o fn, = fv = <t>{D'* < D') o /d- .
Since the right-hand square in diagram 1.68 is a puUback, those relations
imply 0(i5o — ^'*) ° / = /^'* ^ required. D
Lemma 1.14.8 With the notation of 1.14.5, consider an object Dq EV
and a functor T: [Dq >s/ such that for every Di < D2 < Dq the
factorization
T{Di) >limD,<D<DMD)
is an epimorphism. In that case, given an exact functor F: si >Ab
and a natural transformation
aDo'^^{rDo,-)^F,
a£>Q can be factored as
^(r(Do),-)-
>colimD<Do^{^{D),-)-
^F
for some a, with sdq the canonical morphism of the colimit.
Proof Let [Dq indicate the set of elements D < Do.By a. final segment
of [Do we mean a subset S C [Dq such that D e S and D < D' < Dq
imply D' e S.
Let us consider the set S of all pairs E, {aD)Des) where
A) iS is a final segment of jDo,
B) an- s/{r{D), -) =^ F, for D G S, constitute a cocone.

80 Abelian categories
We give <S the structure of a poset by considering
{S,{aD)D€s) < {S\{a^D)Des)
when S C S^ and a^, = a^ for each D e S. S is obviously inductive and
by Zorn's lemma, we choose a maximal element (S, {aD)Des) in it. It
suffices now to prove that S = [Do, from which the unique factorization
a through the colimit.
If 5 7^ IjDo, there exists Di < Do, Di ^ S. Since the segment [Di^Dq]
is finite (see 1.14.6), there exists D2 G [Di.Dq] such that D2 ^ S but
for each D2 < D < Dq, D e S. By the Yoneda lemma (see 1.3.8), the
natural transformations ac'- j/(r(J9), —) ^ F ior D2 < D < Do
correspond with a compatible family of elements a^ e FT{D). Since F is
left exact and there are finitely many such J9's, this compatible family
corresponds with an element a G F(limr(J9)). Since the factorization
/3: T{D2) > limr(J9) is an epimorphism by assumption and since the
functor F is exact, the morphism F(/3): FT{D2) >F{limT{D)) is sur-
jective. Therefore we find aD2 G FT{D2) such that F{aD2) — ^- ^^ P^^"
ticular, composing with the projections, we find FT{D2 < D){aD2) = cld
for each D2 < D < Dq. But aD2 corresponds by the Yoneda lemma with
a natural transformation a£J- si{TD2, —) =^ F such that
ao = OiD20 s^{T{D2 < D),-).
It remains to observe that, due to the choice oi D2, Si) {D2} is final in
i^o. Thus \S^{D2},{oLD)r)^-si^{D2}) extends strictly (^,(aD)^e5),
which contradicts its maximality. D
We are now ready to prove the final result of this chapter.
Theorem 1.14.9 (The full and faithful embedding theorem)
Every small abelian category si has a full, faithful, and exact embedding
in a category Mod^ of modules over a ring R.
Proof We use the notation of 1.14.5.
Step 1: construction of the embedding
We consider the embedding U: si >Ab of 1.14.5 and the abelian
group Nat(C/, C/) of natural transformations on G (see 1.3.2). Taking the
composition of natural transformations as a multiplication, the group
R = Nat(C/, G) becomes a ring, by preadditivity of si.
Given an object A ^ si^ C^(^) is naturally provided with a scalar
multiplication
/?xC/(A) >U{A), (r,x)y-^rA{x),

1.14 The embedding theorem
UD <t>{D < Sa)
<D):
7r(D < D')
<D'):
vd
Ud'
t<i>{Dy
<t>{D < D')
-^> A
u
-14>{D') » A
VD' ^ 4>{D' < 6a)
Diagram 1.69
81
Since va is a group homomorphism and the addition on i? = Nat(C/, U)
is defined pointwise, this is an i?-module structure.
Given an arrow /: A >B, the i?-linearity of U{f) reduces to the
naturality oi r e R:
rB{U{f)){x) = {U{f))rA{x)
where again x G U{A).
Therefore U factors through Mod^ via a functor V: si -
^Modi?.
Since U is faithful, V is faithful. Since limits and colimits are computed
in the same way in Mod^ and in Ab, the exactness of G implies that of
V (see 1.11.12). It remains to prove that V is full.
Sie'p 2: canonical presentation of representable functors
Given an object Ae^ and the corresponding terminal epimorphism
Pa'- a >0, we consider the unique element * e Dq and the index
a + 1 corresponding with the pair (*,pa) in the construction of Pi (see
1.14.5). We obtain Vq'^^ by "duplicating" the initial segment i* C P^,
which is Pq itself. Let us write 6a for the new copy of * G Po which is
so introduced. One has 0Ea) = A and 0F^ < *) = Pa-
For every D < 6a consider diagram 1.69 where (ix£), vd) is the kernel
pair of the morphism 0(J9 < 6a)' H D < D^ < 6a^ consider in the same
way the kernel pair {ud'.vd') of 0(J9' < 6a)' The right-hand square of
diagram 1.69 is commutative, from which we deduce the existence of a
factorization 7r{D < D') making tt: [6a >^ a functor.
Since (j){D < 6a) is an epimorphism, it is a cokernel (see 1.4.1) and
thus the coequalizer of its kernel pair (wd?^d); see 2.5.7, volume 1. But
the contravariant Yoneda embedding transforms coequalizers in
equalizers (see 2.9.5 and the fact that equalizers in Ab are computed as in

82
Abelian categories
7r{Di) ^> • >0(Di)
m
-» Um ttD > lim 0 J9
(l>{Di)—j^>lim(l>D ^—> A
Diagram 1.70
Set). Thus we have an equalizer diagram
^i^D^-)^MD^M^^^A^.y
Let us compute the colimit on J9 < 6^ of each piece of this diagram:
pt
'.U^
w
-<^(A,-).
The colimit of the central terms is indeed G, since [6a is an initial part
of P; see 2.11.2, volume 1. The colimit of the right-hand terms is the
filtered colimit of the constant diagram on s/{A^ —), so it is obviously
j/(i4, —). Since filtered colimits commute with equalizers in Ab, one has
w = Ker{u,v); see 2.13.6, volume 1.
Step 3: the functor tt satisfies the assumptions of 1.14-8
Consider Di < D2 < Sa- By interchange of limits (see 2.12.1,
volume 2) the following diagram is a kernel pair:
limix£)
tlimDi<D<D2 0(^)-
/
-^A
liniD^<D<D2'^{Dy.
Once again, we have used the obvious fact that the limit of the constant
diagram on A is just A, because the indexing diagram is connected (see
2.6.7.e, volume 1).
Now it suffices to consider diagram 1.70 where all the squares are
pullbacks and the factorization m: (t>{Di) >lim£>i<£><£>2(/>(£)) is epi-
morphic by 1.14.7. The composite Z o m is just (j>{Di < 6a) so that the
outer pullback in diagram 1.70 is the one defining 7r(jDi). Since m is

1.14 The embedding theorem
t
( ; \
83
.•••■^(/,-)
Diagram 1.71
uo C
> U
Diagram 1.72
an epimorphism, so are the four sides of the left upper square, which
concludes the argument.
Step 4' V is full
Choose now A^B e^ and a group homomorphism (p: U{A) >U{B)
which is i?-linear for the structures defined in step 1. The monomorphism
w: s/{A^ —) >U of step 2 corresponds by the Yoneda lemma to some
"generic" element a = WAi^A) ^ U{A). Let us consider the element
(p{a) e U{B) which, by the Yoneda lemma, corresponds to a
natural transformation C: j/(B,-) =^ U defined by Pdg) = U{g)[(p{a)).
Refering to diagram 1.71, we shall prove that u o /3 = v o /3^ from
which there exists a unique factorization through w = Ker(ix, t;). By
the Yoneda lemma, this factorization will have the form j/(/, —) for
some /: A >B.
Let us suppose uoP^vof3. C corresponds to (p{a) e U{B) via
the Yoneda lemma and ^{a) G U{B) = colim£><5^j/@J9, B) can be
represented by some morphism x: (j>{D) >B with D < Sb- H uo C ^^
V o i3, one must have x oud t^ x ovd- In diagram 1.72, consider the
monomorphism w^'. j/Gr(jD), —) ) >[/ obtained in step 2, starting with
7r(jD) as object A. Because of step 3 and lemma 1.14.8, there exists a

84
Abelian categories
r.
'■•. r
X
Sd
Diagram 1.73
factorization t: P >U such that to sa = w^. In particular
touof3 = w^o j^{x o ix£), —) z^w' o si(x o vd^ —) = tovo/3
since w^ is a monomorphism. But now tou and tov are elements of R
and given C E s/^ g E s/{B^ C) one would have, since ip and U{g) are
iZ-linear,
{tc ouco f3c){g) = {to u)c o U{g){ip{a))
= U{g)o(po{tou)A{a)
= U{g) oipotAOUAOwa{Ia)
= U{g) oipotAOVAO wa{Ia)
= U{g)oipo{tov)A{a)
= {tov)coU{g){^{a))
= {tcovcof3c){g).
Finally one would have touo/3 = tovo/3^ which is a contradiction.
So we have indeed obtained a morphism /: A >B and it remains to
prove that U{f) = (p. Given any element x G U{A)^ we consider in
diagram 1.73 the corresponding natural transformation x- ^{A, —) >U
obtained from the Yoneda lemma. Let us consider also the
monomorphism w of step 1, which corresponds to some element of U{A)
represented by a morphism y: (t>{D) >j/. Since sd o ^{y-, —) = w and w
is a monomorphism, s/{y^ —) is a monomorphism as well. This implies
that given p, q: A ] B in j/.
J3^B/,-)oj/(p,-) =
-)<
,-)=^p = g
or in other words poy = qoy^p — q. Thus y is an epimorphism in j/
and, since U is exact, U{y): U{<t>{D)) >U{A) is an epimorphism in
Ab, thus a surjection. ChcKjee z € U(<f>{D)) such that U{y){z) = x and
consider the correspondiiig natural transformaticm ^: «s/(^(I>), ~) =^

1.14 The embedding theorem 85
U given by the Yoneda lemma. Since U{y){z) = a;, one has also ^ o
j/(y^—) = ^. By lemma 1.14.7, U satisfies the conditions of lemma
1.14.8 so that we obtain r: U ^ U such that r o 5£> = ^. One has also
row = rosDO ^{y^ —)= i^ «^B/? —) = X-
But since ip is i?-linear and r E R^ we obtain
^W = ^(xaAa))
= (p{rAOWA{lA))
= rBOip{wA{lA))
= rBO (p{a)
= Tb O PsiiB)
= rBOWBO si{f, B){\b)
= XB{f)
= U{f){x).
This concludes the proof. D
Let us observe that since the embedding V: si >Modi? constructed
in 1.14.9 is full and faithful, it reflects isomorphisms (see 1.9.5, volume 1).
Since V is exact, V reflects finite limits and finite colimits (see 2.9.7,
volume 1) thus in particular monomorphisms, epimorphisms and as a
consequence the construction of images. Therefore V reflects exact
sequences. Thus to prove a statement about exact sequences for all small
abelian categories j/, it suffices to prove it for all categories of modules.
The case of arbitrary abelian categories si can generally be reduced
to the case of small ones. Indeed if fi is a diagram of objects and arrows
in j/, let us consider the sequence
where ^o is the full subcategory of si generated by B and ^n+i is
the full subcategory of si generated by the limits and colimits of all
finite diagrams in ^n (choose one limit and one colimit for each finite
diagram). Putting ^ = Un€N^n, ^ is stable in si under finite limits
and finite colimits, thus in particular 0i is an abelian subcategory of si
(see 1.5.7). And when the diagram B is small (which is generally the case
in the applications), the abelian category 3i is still small and so theorem
1.14.9 can be applied to ^, in which the whole problem takes place.
Theorem 1.14.9 can also be seen as a way to introduce diagram
chasing on a small abelian category si (see section 1.9). Indeed, given an

86 Abelian categories
object A e s/^ define a pseudo-element ae*A to be an actual element
a G V{A). The action of a morphism /: A >B on pseudo-elements
is just given by /(a)=defV'(/)(a). Since V preserves and reflects
exact sequences, proposition 1.9.4 is certainly vahd for this new notion of
pseudo-element. But moreover, since V is full and faithful:
A) it is now possible to prove the equality of two morphisms using
pseudo-elements:
yL9:A %B / = ^ iff VaG*A /(a) =* ^(a);
B) it is now possible to construct a morphism /: A >B by describing
its action on pseudo-elements.
Moreover let us observe that since V preserves and reflects puUbacks,
1.9.5 now becomes an equivalence.
1.15 Exercises
1.15.1 In the category Gr of groups, the zero group is a zero object.
Every epimorphism is the cokernel of its kernel. Every monomorphism
is an equalizer, but only the normal subgroups are kernels.
1.15.2 Consider the category Set^ of sets with a base point: an object is
a pair (A, a), with A a set and a G A; a morphism /: (A, a) >{B^b)
is a mapping /: A >B such that /(a) = b. The singleton is a zero
object. Epimorphisms are just surjections. /: (A,a) >{B^b) is a co-
kernel if and only ii f: A\ f~^{b) >B \ {b} is bijective. Prove that
proposition 1.2.3 holds in Set^.
1.15.3 Consider the matrix notation described in section 1.2. Prove that
the sum of two morphisms corresponds with the sum of matrices and
the composite of two morphisms with the product of their matrices.
1.15.4 Consider a field K and an infinite-dimensional iC-vector-space
V, The ring R = EndK(V, V) of K-linear-endomorphisms of F, seen as
preadditive category with a single object, has biproducts but no zero
object.
1.15.5 Given an abelian category j/, consider the category Ar(j/) of
arrows of J2^; see 1.2.7.C, volume 1. Observe that Ar(j/) is abelian and
prove the existence of an exact functor /: Ar(j2/) >j/ mapping an
object /; A >B of Ar(ja/) to its image factorization.

1.15 Exercises 87
1.15.6 In the category Men of abelian monoids, consider the monoid
(Z, +) of integers and its two subobjects (N, +) and (—N, +) of natural
numbers and negative numbers. The union of those two subobjects is
(Z, +), but it is not effective.
1.15.7 By applying 1.8.5.C, prove that epimorphisms are not stable
under puUback in the category Ha us of Hausdorff spaces and continuous
mappings.
1.15.8 By applying 2.2.4.h, prove that monomorphisms are not stable
under pushouts in the category Rng of rings.
1.15.9 Applying the adjoint functor theorem (see 3.3.3, volume 1), prove
that for a small abelian category j/, the category Lex(j/, Ab) of left exact
functors is reflective in the category Add(j/, Ab) of additive functors. (In
fact, this reflection is a localization, but this is much harder to prove.)
1.15.10 Produce a counterexample showing that the following "nine
lemma" is false: suppose five rows and columns are exact in diagram
1.14, then the remaining row or column is exact as well.
1.15.11 Given f = g o h in an abelian category, prove the existence of
an exact sequence
0 >Kerh >Ker/ ^Ker^f- • •
>Coker/i >Coker/ >Coker^ >0.
1.15.12 Prove the following version of the nine lemma. Consider
diagram 1.14 where the middle row and the middle column are exact, the
square A) is a puUback, the square B) is a pushout and e o /3, i/ o9 are
image factorizations. Then all the rows and columns are exact.
1.15.13 Consider an abelian category s/ and the category Sex{s/) of
short exact sequences of j/: an object is an exact sequence and a mor-
phism is a triple /, ^, /i of morphisms making the squares of diagram 1.74
commutative. Prove that this category has biproducts, kernels and co-
kernels. Prove that when g and h are monomorphisms in j/, (/, g^ h)
is a monomorphism in Sex(j/). Prove that when {f,g,h) is a kernel in
Sex(j/), /, ^, h are monomorphisms in j/ and the left-hand square is a
puUback. Produce a counterexample showing that in general Sex(j/) is
not abelian. [Hint: construct the kernel of (/, g^ h) as the following exact
sequence:
0 >Ker/ >Ker^ >X >0.]

88
0
Abelian categories
-> A > B > C
f
-> A!
-> B'
Diagram 1.74
-^ a
-> 0
^ 0
1.15.14 Prove the snake lemma using the full embedding theorem 1.12.9,
i.e., prove the existence of the diagonal morphism by just working with
"elements".
1.15.15 Consider the category of abelian groups and the torsion theory
defined at the beginning of section 1.12. Prove that the corresponding
closure operation is given by
^ ={aeA \ 3n G N, n^O, na e S}
for a subgroup S C A.
1.15.16 On the category of abehan groups, consider the closure
operation defined as in 1.15.15 for all subgroups S C A with A ^ Z; for a
subgroup 5 C Z, define S = S. Prove that this closure operation
satisfies the three axioms indicated at the end of section 1.12, but it is not
induced by a torsion theory.

2
Regular categories
Abelian categories are additive (see 1.6.4), which excludes many
interesting situations in which, nevertheless, several "exactness properties" of
abelian categories are still valid, like the existence of images (see 1.5.5).
The notion of a regular category recaptures many "exactness properties"
of abelian categories, but avoids requiring additivity. For example, the
category of sets and most "algebraic-like" categories are regular.
Convention: as a matter of convention, in the present chapter, the
symbol » will denote a regular epimorphism; no special notation
will be used for ordinary epimorphisms.
2.1 Exactness properties of regular categories
We recall that an epimorphism is regular when it can be written as the
coequalizer of some pair of morphisms (see 4.3.1, volume 1). Regular
epimorphisms and kernel pairs are closely related via the properties:
• if a regular epimorphism has a kernel pair, it is the coequalizer of
that kernel pair (see 2.5.7, volume 1);
• if a kernel pair has a coequalizer, it is the kernel pair of that regular
epimorphism (see 2.5.8, volume 1).
The reader will also remember from 4.3.6 of volume 1 that
• every regular epimorphism is strong.
The properties of strong epimorphisms have been studied in section 4.3,
volume 1.
There exist in the literature many different definitions of regular
categories, which are all equivalent under the assumptions that finite limits
89

90
Regular categories
9
D
-^ B
-> C
f
Diagram 2.1
and coequalizers exist. We choose here a somehow "weakest" possible
definition. Requesting the existence of more finite limits in the
definition of a regular category is essentially a matter of personal taste or
convenience; requiring the existence of all coequalizers is less innocuous:
for example arbitrary coequalizers are not preserved by exact functors
(see 2.3.5 and 2.7.4), by pulling back in a regular category (see remark
after 2.1.1), by forgetful functors in algebraic situations (see 3.9.1 and
4.4.4), and so on.
Definition 2.1.1 A category ^ is regular when it satisfies the following
conditions:
A) every arrow has a kernel pair;
B) every kernel pair has a coequalizer;
C) the puUback of a regular epimorphism along any morphism exists
and is again a regular epimorphism.
The third axiom means that in diagram 2.1, if / and p are given with
p a regular epimorphism, then the puUback (q^g) of (/,p) exists and q
is a regular epimorphism.
It is probably useful to emphasize the fact that the third axiom in
2,1.1 requires the preservation of regular epimorphisms under puUbacks,
not the preservation of coequalizers (see exercise 2.9.1).
Lemma 2.1.2 Consider a regular epimorphism /: A »B and an
arbitrary morphism g: B >C. In these conditions the factorization
fxcf: AxcA >BxcB
exists and is an epimorphism.
Proof The puUback BxcB of the pair {g, g) is just the kernel pair of g,
thus it exists. Since / is a regular epimorphism, the three other partial
puUbacks involved in diagram 2.2 exist, yielding regular epimorphisms
d, e, i^j. Then fxcf = doi = eoj, wliich is an epimorphism as composite
of two (regular) epimorphisms (see 1.8.2, volume 1). D

2.1 Exactness properties of regular categories
AxcA—^>BxcA—-—^ A
f
AxcB—^>BxcA ^—> B
91
R
-^> B
t I
-> C
f ■ - 9
Diagram 2.2
zz:^ A —^—^ B
Diagram 2.3
Theorem 2.1.3 In a regular category, every morphism factors as a
regular epimorphism followed by a monomorphism and this factorization
is unique up to isomorphism.
Proof Consider a morphism /, its kernel pair (ix, v) and the coequalizer
p = Coker (ix,t^) of that kernel pair, as in diagram 2.3. Since fou = fov^
we get a unique factorization i through the coequalizer, yielding / = iop.
It remains to prove that i is a monomorphism.
Let (r,s) be the kernel pair of i. Since iopou = iopov^ there exists
a unique morphism q such that roq = pou^ soq= pov. Applying 2.1.2
to the regular epimorphism p and the morphism i, we observe that
P = AxbA, R = IxbI, q = P^BP,
thus q is an epimorphism. Then roq^pou = pov = soq implies r = s
and i is indeed a monomorphism (see 2.5.6, volume 1). The uniqueness
of the factorization is attested by 4.4.5, volume 1. D

92 Regular categories
As usual, the morphism i in 2.1.3 (or sometimes just its domain /) is
called the image of /.
Proposition 2.1.4 In a regular category, the following conditions are
equivalent:
A) f is a regular epimorphism;
B) f is a strong epimorphism.
Proof A) =^ B) has been proved in 4.3.6, volume 1. Conversely a
strong epimorphism / factors as f = iop with i a monomorphism and
p a regular epimorphism. But then i is both a monomorphism and a
strong epimorphism, thus an isomorphism (see 4.3.6, volume 1). D
Corollary 2.1.5 In a regular category:
A) the composite of two regular epimorphisms is a regular epimorphism;
B) if a composite f og is a regular epimorphism, f is a regular
epimorphism;
C) a morphism which is both a regular epimorphism and a
monomorphism is an isomorphism.
Proof Via 2.1.4, this is just 4.3.6, volume 1. D
2.2 Definition in terms of strong epimorphisms
In a regular category, regular epimorphisms coincide with strong
epimorphisms (see 2.1.4). Therefore one can expect a definition of a regular
category in terms of strong epimorphisms.
In definition 2.1.1, the second axiom asserts the existence of "enough"
regular epimorphisms, while the third axiom is an "exactness" property
of regular epimorphisms. If the third axiom makes perfectly good sense
with "regular" replaced by "strong", the second axiom must be put into
an equivalent form to allow an analogous translation.
Proposition 2.2.1 A category ^ is regular when it satisfies the
following conditions:
A) every arrow has a kernel pair;
B) every arrow f can be factored as f = iop with i a monomorphism
and p a regular epimorphism;
C) the pullback of a regular epimorphism along any morphism exists
and is a regular epimorphism.

2.2 Definition in terms of strong epimorphisms 93
u
f
V
t A —» B
9
■■.h
11
Pb
C < p^ BxC
Diagram 2.4
Proof Theorem 2.1.3 proves the necessary condition. Conversely,
consider an arrow /, its kernel pair (ix, v) and its mono-regular-epi
factorization f = iop. Since i is a monomorphism, (ix, v) is still the kernel pair
of p and since p is a regular epimorphism, p is the coequalizer of (ix, v);
see 2.5.7, volume 1. D
The conditions of 2.2.1 can now be stated for strong monomorphisms,
yielding the following result.
Proposition 2.2.2 Let ^ be a category with finite limits. The category
^ is regular if and only if it satisfies the following conditions:
A) every arrow f can be factored as f = iop with i a monomorphism
and p a strong epimorphism;
B) the puUback of a strong epimorphism along any morphism is again
a strong epimorphism.
Proof The necessity of conditions A) and B) follows from 2.1.5 and
2.2.1. Conversely, assuming the conditions of 2.2.2, it suffices by 2.2.1 to
prove the coincidence between strong and regular epimorphisms. To do
this, let us consider a strong epimorphism /: A >B, the kernel pair
(ix, v) of / and a morphism g such that g o u = g o v; we shall prove the
existence of w: B >C such that g = w o f (such a, w is necessarily
unique since / is an epimorphism). We consider the product of B, C and
the unique factorization h: A >B x C as in diagram 2.4, such that
Pb ^ h = f ^ PC o h = g. The morphism h can be factored as h = i o p
with i a monomorphism and p a strong epimorphism. We shall prove
that Pb oiis an isomorphism and w = pc oio [p^ oi)~^ is the required
factorization.
Let us consider diagram 2.5, where all the individual squares are pull-
backs. Since pBoiop = pBoh = f,we can identify the global puUback
with the kernel pair of /, yielding P' = -P, xom = % yon = v. Since p
is a strong epimorphism, so are t, ^, m, n; r, s are strong epimorphisms

94
Regular categories
u\
r
p'
m
X
^
n
^> Y
-> Q
■^> A
^> /
Vb°i
V A —^^> I ^^^ B ^
J
f
Diagram 2.5
as well as parts of a kernel pair, but this does not play any role here. By
commutativity of the diagram, we have immediately
PBoiorotom = pBoiosotom.
On the other hand
Pcoiorotom = pcoiopoxom
= pcohoxom
= gou
= gov
= pcohoyon
= Pcoiopoyon
= Pcoiosoqon
= pcoiosotom.
By definition of a product, this yields iorotom = iosotom^ thus
rotom^ sotom since i is a monomorphism and r = s since ^, m are
epimorphisms. This already proves that pb oi'is b. monomorphism (see
2.5.6, volume 1). But since / = (jpsoi) op and / is a strong epimorphism,
pB^i is a strong epimorphism as well and finally p^ o^ is an isomorphism
(see 4.3.6, volume 1).

2.3 Exact sequences 95
So we put w = pc^io {pb oi)~^. It is straightforward to observe that
wo f =pcoio{pBO i)~^ o /
= PC oio {pb oi)~^ opB oiop
= PC oiop
= 9
On the other hand we have noticed already that such a factorization w
is unique since / is an epimorphism. This proves that / = Coker{u,v).
In particular, every strong epimorphism is regular and thus regular
epimorphisms coincide with strong epimorphisms (see 4.3.6, volume 1). D
2.3 Exact sequences
Definition 2.3.1 By an exact sequence in a regular category we mean
a diagram
U n
P U ^ >B
V
where (ix, v) is the kernel pair of f and f is the coequalizer of (ix, v).
Observe that in definition 2.3.1, u^v are regular epimorphisms since
(for example) / is a regular epimorphism.
We shall observe in 2.4.1 that an abehan category is always regular.
The relation of 2.3.1 with the notion of exact sequence in an abelian
category is given by the following proposition.
Proposition 2.3.2 Let ^ be an abelian category The following
conditions are equivalent:
u
A) P I A ^ ^ B is an exact sequence in the sense of 2.3.1;
B) the sequence
(:)
0 >P ^ ^ ^ yA^A ^^' -^^ )g >0
is a short exact sequence in the sense of 1.8.6.
Proof For morphisms x^y: X > A^ the relation fox = f o y is
equivalent to (/, —/) o (^) = 0; this proves that (ix, v) is the kernel pair of
/ iff (^) is the kernel of (/, -/). Now assuming A), / is an epimorphism
and the sequence in B) is a short exact sequence (see 1.8.5). Conversely
assuming B), (/, -/) = /o A^, -1^) so that / is an epimorphism, thus
a regular epimorphism since ^ is abelian; one concludes the proof by
2.5.7, volume 1. D

96 Regular categories
u' f
P' ] A' > B'
k
h
u
t A p^ B
V J
Diagram 2.6
Proposition 2.3.3 In a regular category, pulling back along any arrow
preserves exact sequences.
Proof We consider the situation of diagram 2.6, where all the individual
squares are puUbacks and {f\u^v) is an exact sequence. Observe that
since fou = fov, their puUbacks with g are the same and by associativity
of puUbacks (see 2.5.9, volume 1); this means the existence of a single
morphism k such that {u\ k) is the puUback of (ix, h) and (t^', k) is the
puUback of {v^ h). We must prove that (/'; ix', v^) is an exact sequence.
An easy "diagram chasing argument" shows that {u' ^v') is the kernel
pair of f. Indeed f ox = f oy implies fohox = gofox = gof'oy =
f o h o y^ from which there is a unique w such that u o w = /i o x,
vow = hoy. This yields 2:1,2:2 such that ix' o 2:1 = x, fc o zi = w
and v' o Z2 = y-, k o Z2 = w. The relations kozi=w = koz2 and
f ou'ozi= f ox = f o'y = f ov'oz2 = f ou'0Z2 imply Zi =2:2, since
the global diagram is a puUback. This yields a morphism 2: = 2:1 = 2:2
such that v! oz — X and v' o z = y. The uniqueness of such a 2: is proved
in the same way.
Now f is regular since / is. The pair {u' ^v') is the kernel pair of f
and f is regular: this implies that f is the coequalizer of {u'^v')\ see
2.5.7, volume 1. D
Proposition 2.3.4 Let ^ be a regular category with binary products.
Given two exact sequences
U n U' ft
J. ^R P' ^ A' J
Za ^- >B, P' \A' ^- ^B',
V
the product sequence ^
UXU\ r p
P X P' \A X A' ^ ^^ >B X B'
• • V X V^
m agam exact
Proof It is just an obvious exercise on puUbacks and products to check
that {u xu'.v X v') is the kernel pair of / x f. By 2.5.7, volume 1, it

2.3 Exact sequences 97
A X A'^^B xA' Bx A'^^B' x A'
PA
Pb
-» B
PA'
Diagram 2.7
Pb'
A' p—» B'
remains to prove that / x /' is a regular epimorphism. Observing that the
squares of diagram 2.7 are puUbacks one concludes that / x 1 and Ix f
are regular epimorphisms. This implies that / x /' = (/ x 1) o A x f)
is also a regular epimorphism (see 2.1.5). D
In 1.11.2, we observed that for an additive functor between abeUan
categories, preservation of short exact sequences did imply preservation
of finite limits and colimits. This was essentially due to the equational
characterization of biproducts, given in 1.2.4, and the close relation
between kernels and equalizers. No such phenomena appear for regular
categories, so that the definition which is generally admitted is the
following one.
—>^ be a functor between regular cate-
Definition 2.3.5 Let F: %
gories ^, S>. The functor F is exact when it preserves:
A) all finite limits which happen to exist in ^;
B) exact sequences.
In most applications, the regular categories one considers have finite
limits. When this is not the case, it is not clear if condition A) in
definition 2.3.5 is pertinent. One could imagine replacing it by a weaker
condition, like the preservation of just those finite limits whose
existence is required in the definition of a regular category, or by a stronger
condition, like being flat (see 6.7.5).
Proposition 2.3.6 Let F:
-^S> bean exact functor between regular
categories ^, S>. The functor F preserves:
A) regular epimorphisms;
B) kernel pairs;
C) coequalizers of kernel pairs;
D) mono-regular-epi factorizations.
n

98 Regular categories
Proposition 2.3.7 Let F: ^ >S> be a functor between regular
categories. The following conditions are equivalent:
A) F is exact;
B) F preserves finite limits and regular epimorphisms.
Proof A) =^ B) by 2.3.6. Conversely, with the notation of 2.3.1, the
regular epimorphism F{q) has a kernel pair (F(ix),F(?;)), thus is its
coequalizer (see 2.5.7, volume 1). D
Proposition 2.3.8 Let F: ^ >S> be an additive functor between
abelian categories ^, S>. The following conditions are equivalent:
A) F is exact in the sense of definition 2.3.5;
B) F is exact in the sense of definition 1.11.1.
Proof B) =^ A) since by 1.11.2, F preserves finite limits and finite
colimits.
Conversely, assuming the conditions of 2.3.5, F preserves kernels, since
it preserves finite limits. It also preserves (regular) epimorphisms by
2.3.6. Thus F is exact by 1.11.4. D
2.4 Examples
Example 2.4.1
An abeHan category is finitely complete and cocomplete (see 1.5.3) and
every epimorphism is regular (see 1.5.7). By the dual of 1.7.6, every
abeUan category is regular.
Example 2.4.2
The category of sets is finitely complete and cocomplete (see 2.8.6,
volume 1). All epimorphisms are strong (see 4.3.10, volume 1): they are the
surjections (see 1.8.5.a, volume 1); monomorphisms are just injections
(see 1.7.8.a, volume 1). Every mapping factors as a surjection followed
by an injection. Moreover the puUback of a surjection is obviously a
surjection. Indeed if diagram 2.8 is a puUback of sets with g a surjection,
AxcB = {{a,b)\aeA,; beB, f{a)=g{b)}.
Given a e A, there exists b e B such that /(a) = g{b)^ just because g
is a surjection. Therefore (a, 6) G AxcB and pA{a^b) = a. This proves
that PA is surjective. So the category of sets is regular by 2.2.2.

2.4 Examples 99
AxcB ^^ ) B
Pa
» C
f
Diagram 2.8
Example 2.4.3
The category of groups is finitely complete and cocomplete (see 2.8.6,
volume 1). All epimorphisms are strong (see 4.3.10, volume 1) and
coincide with the surjections (see 1.8.5.d, volume 1); monomorphisms are
just injections (see 1.7.8.C, volume 1). The pullback of a surjection is a
surjection and every homomorphism factors as a surjective one followed
by an injective one. By 2.2.2, the category of groups is regular.
Example 2.4.4
The kernel pair of a monomorphism is just the identity pair (see 2.5.6,
volume 1) and, of course, the coequalizer of the identity pair is just the
identity. With 2.8.7, volume 1, in mind, this proves that in a category
^ where every arrow is a monomorphism, the regular epimorphisms are
exactly the isomorphisms. Therefore a category where every arrow is a
monomorphism is necessarily regular. In particular, every poset, seen as
a category, is regular (see 1.2.6.b, volume 1). The category of fields is
also regular, since every homomorphism of fields is injective.
Counterexample 2.4.5
The category Top of topological spaces and continuous mappings is not
regular. Indeed, the strong epimorphisms are just the quotient maps
/: A >B, i.e. the surjections / where B is provided with the
corresponding quotient topology (see 4.3.10.b, volume 1). But quotient maps
are not stable under puUbacks, so that Top is not regular. Here is an
elementary counterexample. Let us put
A = {a, 6, c, d} with {a, b} open,
S = {Z, m, n} with {Z, m} open,
C = {a;, y, z} with the indiscrete topology.
We define /: A >C, g: B >C by
/(a)=x, f{b)=y = f{c), m=z,

100 Regular categories
9{l) = x, g{m) = z = g{n).
Now / is surjective and no subset of C has {a, h] as inverse image; thus
/ is a quotient map. The product Ax B has a single non-trivial open
subset, namely {a, h) x {Z, m}. The puUback of /, g is thus given by
P = {(a, Z), (d, m), (d, n)} with {(a, Z)} open.
The projection pc'- P ^C' is not a quotient map since p^^(Z)={(a, Z)}
is open while {Z} is not.
Counterexample 2.4.6
The category Cat of small categories and functors is not regular. Indeed,
strong and regular epimorphisms do not coincide in this category Cat
(see 4.5.17.h, volume 1 and 2.1.4).
Example 2.4.7
Consider a regular category ^ and a small category Q). The category of
functors and natural trasformations Fun(^, ^) is again regular. Indeed,
the considerations of 2.15.1, volume 1, indicate that kernel pairs and
their coequalizers can be constructed pointwise in Fun(®,^) since they
exist in ^. Now if a: F =^ G is a regular epimorphism in Fun(®,^), it
is the coequalizer in Fun(^, ^) of its kernel pair (see 2.5.7, volume 1);
therefore for every object J9 G ^, a£> is in ^ the coequaUzer of its kernel
pair, yielding an exact sequence in ^. Since exact sequences in ^ are
preserved under puUing back, one concludes immediately that puUing
back a in Fun(^,^) yields another regular epimorphism.
Example 2.4.8
Choosing a small category ^, the category Fun(^, Set) of set valued
functors on S) is regular (see 2.4.2 and 2.4.7). This example is a "generic"
one in the sense that every small regular category ^ admits a full exact
embedding <
^ >Fun (^,Set)
in some category of set-valued functors. The proof of this theorem is
essentially a more sophisticated version of the method used in section
1.14 (see Barr, Exact categories). It yields as a corollary a metatheorem
asserting that to prove, in a regular category, a property of exact
sequences, it suffices to prove it in a category of set valued functors. Now
sinc^ exact sequences in Fun(^, Set) are constructed pointwise, this often
reduces the problan to giving a proof in the category of sets. In section

2.5 Equivalence relations 101
2.7 we shall prove an alternative metatheorem using the techniques of
topos theory (see chapter 6, volume 3).
Example 2.4.9
If ^ is a regular category, for every object C G ^ the category ^/C
(see 1.2.7.a, volume 1) is regular. Indeed puUbacks and coequalizers in
^/C are easily seen to be computed as in ^ (see 2.16.3, volume 1), from
which the result follows immediately.
Observe that when ^ is abelian (see 2.4.1), ^/C does not need to
be so since it is in general not additive (see 1.6.4). Indeed given two
morphisms /, g: (X, x) ^ (Y, y) in ^/C, one has yof = x^yog = x
from which yo{f-\-g) = x-\-x... and x -\- x ^ x except when x = 0.
Example 2.4.10
In chapters 3, 4, we shall prove that the algebraic and monadic categories
over Set are regular (see 3.5.4 and 4.3.5).
2.5 Equivalence relations
A relation on a set A is a subset RC Ax A. But giving the embedding
r: R> >A x A is equivalent to giving the two composites
Pi o r,p2 o r: R> >A x A '^ A.
We prefer this second formulation since it does not refer to the existence
of cartesian products. The injectivity of r can immediately be translated
by the fact that (pi o r,p2 o r) is a monomorphic family (see 4.8.5,
volume 1), i.e. that two arrows a;, y: X > R are equal iSpiorox = pioroy
(i = l,2).
Definition 2.5.1 By a relation on an object A of a category ^, we mean
an object R E^ together with a monomorphic pair of arrows
ri,r2: R ]A
(i.e. given arrows a;, y: X I R^ x = y iffriox — rioy and r20x = r2oy).
For every object X e^ we write
Rx = {{riox,r2 0x)\xe ^(X,R)}
for the corresponding relation (in the usual sense) generated by R on
the set ^(X, A).
Given a relation (i?, ri, r2) on an object AG^oi some category, it is
now possible to require classical properties on the various relations Rx
on the sets ^(X, A).

102 Regular categories
RxaR—^^—> R
Pi
n
R rT^ ^
Diagram 2.9
Definition 2.5.2 By an equivalence relation on an object A of a
category ^, we mean a relation (i?, ri,r2) on A such that, for every object
X E ^, the corresponding relation Rx on the set ^(X, A) is an
equivalence relation. More generally the relation R is reflexive (respectively
transitive, symmetric, antisymmetric, ...) when each relation Rx is.
In the category of sets, given an equivalence relation RC Ax A^ one
can perform the quotient of A by i?, yielding a diagram
R ]a 2—^A/R,
The coequalizer of ri, r2 is the quotient of A by the equivalence relation
generated by the pairs {ri{x)^r2{x))^x G i?,..., i.e. the quotient of A
by i?, which is q. On the other hand q{a) = q{a') iff (a, a') G i?, which
indicates that (ri,r2) is the kernel pair of q.
Definition 2.5.3 An equivalence relation (i?, ri,r2) on an object A of
a category ^ is effective when the coequalizer q of (ri,r2) exists and
(ri, r2) is the kernel pair of q.
Proposition 2.5.4 Let ^ be a category admitting puUbacks of strong
epimorphisms, A relation (i?, ri, r2) on an object A e^ is an equivalence
relation precisely when there exist:
A) a morphism 6: A >R such that noS = 1a, r2o6 = 1a;
B) a morphism a: R >R such that noa = r2, r2oa = ri;
C) amorphismr: RxaR >Rsuch thatrior = riopi, r20T = r20p2,
where the puUback is that of diagram 2.9.
Such morphisms 6, cr, r are necessarily unique.
Proof The reflexivity of the relation (i?, ri,r2) implies that given the
pair Aa, 1a)- A > A, there exists a morphism 6: A >R such that
no S = 1a = r2 0 S, Conversely, if given the relation (i?, ri, r2) such a
morphism S exists, then for every arrow x: X >A one has
X = rio6 o X, X = r2 o 5 o X,

2.5 Equivalence relations 103
which proves that (a;, x) G Rx and thus R is reflexive.
The pair (ri,r2): R IA is obviously in Rr^ since r*i = ri o 1^ and
7*2 = r2 o 1^. By symmetry of (i?, r*i, r2), the pair (r2, r*i): /? ] A is in
i?i^, yielding a morphism a: R >R such that ri o cr = r2, r*2 o cr = ri.
Conversely, if given the relation (i?, T*i,r2) such a morphism a exists,
then for every pair of arrows x^y: X ^A in Rx^ one has a morphism
z: X >R such that ri o z = x^ r2 o z = y; as a consequence,
rioa o z = r20 z = y, r2 o a o z = ri o z = x^
and the pair (y, x) is in Rx as well, proving the symmetry of R.
Observe that the definitions of 6 and a were independent of the
existence of finite limits. This is no longer true for the morphism r
"representing" the transitivity of (i?, ri,r2). Let us consider the puUback of
diagram 2.9, which exists since the relations ri o 6 = Ia, ^2 o ^ = 1a
imply that ri, r2 are strong epimorphisms (see 4.3.6, volume 1).
Intuitively, RxaR represents the pairs ((ai,a2), @2,03)) where ai ^ 02 and
a2 « 03. Considering the diagrams
ri ri
RxaR ^—^ff ]A^ RxaR ^—^/? ]A.
r*2 T*2
We conclude that {riopi^r20pi) and (r*iop2,7*20/92) are in R^rxj^r)- Since
7*2 o/9i = ri 0/92, this implies that (ri 0/91, r2 0/92) is in R(rxj^r), yielding
an arrow r: RxaR >R such that rior = riopi and r2 or = r2 op2.
Conversely, suppose we are given a relation (J?, ri, r2) with the property
that such a morphism r exists. Given three arrows a;, y, z: X > A with
(a;, y) e Rx and (y, z) e Rx^ we get two morphisms ix, t^: A" ] R such
that ri o u = X, r2 o u = y^ ri o V = y^ r2 o V = z. Prom the relation
n ot? = r2 oix we get a morphism w: X >RxaR such that piow = u^
P20W = V. Finally one has
X = riou = rio piow = riOT ow^
z = r2 o V = r2 o p2 o w = r2 o T o w^
so that (a;, z) e Rx and R is transitive.
It remains to observe that the morphisms 6, cr, r with the indicated
properties are unique, because (r*i,r2) is monomorphic. D
Finally when products exist, equivalence relations admit a more usual
description.
Proposition 2.5.5 Let ^ be a category with Rnite limits. An
equivalence relation on an object A of si can equivalently be dehned as a
subobject r: R> >A x A such that:

104 Regular categories
A) the diagonal A a'- A) >A x A factors through r;
B) with Ea' A X A >A x A as the symmetry on A, i.e. the unique
morphism such that pi o Ha = P2 ^^d p2 o T>a = Pi, then T>a o t
factors through r;
C) consider vi = pior, r2 = P2^f^ and the pullback of diagram 2.9; the
morphism
(''''^']:RxaR >AxA
factors through r.
Proof Given an equivalence relation ri,r2: R > A on an object A^
we get a corresponding factorization
r: R >AxA, pior = ri, P2or = r2,
which is a monomorphism, since the pair (T*i,r2) is monomorphic.
Conversely given a monomorphism r: R> >A x A^ one defines ri,r2 as in
the statement and this family is monomorphic because r is.
Conditions A), B), C) of 2.5.4 for ri,r2 are just conditions A), B),
C) of 2.5.5 for r. D
Examples 2.5.6
2.5.6.a As we observed before giving definition 2.5.3, equivalence
relations are effective in the category of sets.
2.5.6.b If ® is a small category, equivalence relations are effective in
the category of set valued functors Fun(®, Set). Indeed, coequalizers and
kernel pairs are constructed pointwise and the result holds in Set.
2.5.6.C In the category Gr of groups, equivalence relations are effective.
Indeed, an equivalence relation ri,r2: RZHXG on a group G is just a
congruence on G and the coequalizer is performed exactly as in the case
of Set (see 2.4.6.d, volume 1).
2.5.6.d In an abelian category, equivalence relations are effective.
Indeed, given an equivalence relation ri,r2: RZHXA^ we can compute its
coequalizer q: A >Q (see 1.5.3) and it remains to prove that (r*i,r2)
is the kernel pair of q. But being an equivalence relation is a property
which can be expressed entirely in term of finite limits (see 2.5.4); thus
by the embedding theorem for abelian categories (see 1.14.9) it suffices
to prove the result in a category of modules. And this is obvious since the
coequalizer of ri,r2 in a category of modules is just the set theoretical
quotient of A by the congruence generated by the pairs {ri{x),r2{x)),
i.e. exactly the set theoretical quotient of A by the equivalence relation

2.6 Exact categories 105
i?, which is already a congruence (which means an equivalence relation
which is a submodule oi A x A).
2.5.6.e In every category ^, the pair A^, 1a)- AZU^Ais obviously an
effective equivalence relation on the object A. When ^ has finite limits,
it corresponds with the diagonal subobject of A^: A> >A x A.
2.5.6.f In the category Top of topological spaces and continuous
mappings, equivalence relations are not effective. Indeed given a topological
space A and an equivalence relation R on A^ one gets an equivalence
relation R> >A xAinAxAhy providing the set R with any topology
stronger than the topology induced by the product topology on Ax A.
On the other hand, the kernel pair of a morphism /: A >B has always
the topology induced by that oi A x A.
2.5.6.g In every category ^ with coequalizers, a kernel pair is always
an effective equivalence relation. The fact it is an equivalence relation is
just obvious and the effectiveness follows from 2.5.8, volume 1.
Let us conclude this section with a warning. In 2.5.5.d we used the
embedding theorem of 1.14.9 to prove the effectiveness of equivalence
relations in an abelian category. By 2.5.5.b, equivalence relations are
effective in every category Fun(®,Set) and, as mentioned in 2.4.8, there
exists for every small regular category ^ a full exact embedding of the
form ^ >Fun(®,Set). Nevertheless, one cannot conclude that
equivalence relations are effective in every regular category (an explicit
counterexample will be produced in 2.6.12). The difference with the abelian
case is that an exact functor in the sense of 2.3.5 preserves coequalizers
of kernel pairs, not arbitrary coequalizers (compare with 1.11.2). And in
the construction developed in 2.5.5.d, the whole problem is precisely to
prove that q (supposing it exists) admits (r*i,r2) as a kernel pair.
2.6 Exact categories
Definition 2.6.1 An exact category is a regular category in which
equivalence relations are effective.
Putting together the considerations of 2.4 and 2.5.6, we get the
following examples of exact categories: every abelian category, the category
of sets, the category of groups, every category Fun(®, Set) of set-valued
functors on a small category ^. In chapters 3 and 4 we shall prove that
algebraic and monadic categories over Set are exact (see 3.5.4 and 4.3.5).

106 Regular categories
In this section, we want to emphasize the fact that exact categories are
exactly the "non-additive version of abelian categories". More precisely,
we want to show that an exact additive category is necessarily abelian.
Lemma 2.6.2 In a non-empty and preadditive regular category ^, the
biproduct A® A exists for every object A.
Proof Consider an arbitrary object A e ^^ the zero map 0: A >A
and its kernel pair u^v: P > A. Given arbitrary morphisms
x^y: XZHXA^ one has 0 o a; = 0 o y, from which there is a unique
morphism z: X >P such that uo z = x^ v o z = y. This proves that
(P^u^v) is the product Ax A. One derives the conclusion by 1.2.4. D
Lemma 2.6.3 A non-empty and preadditive regular category ^ has a
zero object.
Proof Choose an arbitrary object A. By lemma 2.6.2, the zero map
0: A >A admits pi,P2- ^ ® A IA as kernel pair. Let q: A >Q
be the coequalizer of pi,p2- Given a morphism x: Q >X one has
xoq = xoqopiosi = x o q o p2 o si = xoqoQ = 0
(see 1.2.4), from which a; = 0, since q is an epimorphism. Thus 0 is the
unique morphism from Q to X and Q is an initial object. One derives
the conclusion by 1.2.3. D
Lemma 2.6.4 A non-empty and preadditive regular category ^ has
biproducts.
Proof Given objects A, B, the morphisms A >0, B >0 to the
zero object (see 2.6.3) are retractions, with zero as a section. Therefore
they are regular epimorphisms (see 6.5.4, volume 1) and the puUback of
those two arrows exists, yielding the product A x B; see 2.8.2, volume 1.
One derives the conclusion by 1.2.4. D
Lemma 2.6.5 A non-empty and preadditive regular category ^ has
kernels.
Proof Take a morphism /: A >B and its kernel pair u^v: P ] A.
The morphism u — v: P >A can be factored asu — v = iop with i
a monomorphism and p a regular epimorphism. We shall prove that i is
the kernel of /.
First / o i o p = / o (n — v) = if ^ u) — {f o v) = 0, which proves
/ o i = 0 since p is an epimorphism. Next if x: X >A is such that

2.6 Exact categories 107
/oa; = 0 = /oO, we get a factorization y: X >P such that uoy = a;,
V oy = 0. This yields
iopoy=(u — v)oy = (uoy) — (voy) = x — 0 = x
and p o 2/ is a factorization of x through i. This factorization is unique
since i is a monomorphism. D
Lemma 2.6.6 A non-empty preadditive regular category ^ is finitely
complete.
Proof By 1.2.8 and 2.6.5, ^ has equaUzers. One derives the conclusion
by 2.6.4, this volume and 2.8.1, volume 1. D
Lemma 2.6.7 In a preadditive category ^, every reflexive relation is
necessarily an equivalence relation.
Proof Consider a reflexive relation si,S2- S I A. Given an object
X G ^, the relation
Sx = {E1 o a;, 52 o x)\ X e ^(X, S)}
on the abeUan group ^(X, A) contains the diagonal, just by assumption.
Since Sx is obviously a subgroup of ^(X, A) x ^(X, A), this reduces the
problem to proving the lemma in the category of abelian groups.
Supposing ^ = Ab, we know already that all the pairs (a, a) belong
to S, thus S is reflexive. Next if (a, b) G S, using the reflexivity we get
F, a) = (a, a) - (a, b) + F, b) e S
proving the symmetry of S. Finally if (a, b) e S and F, c) G 5, again
using the reflexivity we obtain
(a, c) = (a, b) - F, b) + F, c) G S,
which proves the transitivity of S. D
Lemma 2.6.8 In an additive exact category ^, every monomorphism
has a cokernel and is the kernel of its cokernel.
Proof Let /: A) >B be a monomorphism. Applying 2.6.4 and using
the matrix notations of 1.2, let us consider the morphism
\0 1b J
A e B >B e B.

108 Regular categories
This is a monomorphism since given morphisms
so that from the relation
we deduce b = b' and {f o a) -\-h = {f o a') -\-b', thus f oa = f oa' and
finally a = a' since / is a monomorphism. Observing moreover that
/ 1b
0 1b
:)=(::)-«■
we conclude that the monomorphism r, seen as a relation on S, contains
the diagonal Ab- Therefore r is an equivalence relation by 2.6.7 and
thus an effective equivalence relation, because ^ is an exact category
(see 2.6.1).
Writing q for the coequalizer of the effective equivalence relation r on
jB, we thus have an exact sequence in ^
A®B Xb 2—^Q.
@,1b)
In particular
^ o / = ^ o (/, 1b) o 5^ = g o @, 1b) o 5a = g o 0 = 0.
Now given a morphism x: B >X such that xo f = 0
Xo (/, 1b) = {xof,x) = @,x) = xo @, 1b)
from which we get a unique factorization z: Q >X such that zoq = x.
This proves that q = Coker/.
Thus / admits g as a cokernel. It remains to prove that / is the kernel
of q. Given y: Y >B such that qoy = 0 = qo0^we find a unique
z: Y >A e B such that (/, 1b) o 2: = y, @, 1b) o 2: = 0. The arrow
z has the form (JJ) for some morphisms u: Y >A, v: Y >B. The
arrow u is the required factorization since
0 = @, 1b) o r M = @ o w) + Ab ov) = v,
2/=(/,1b)o ( M =^ if ou) + {1b ov) = {f ou) + {1b oO) = f ou.

2.6 Exact categories 109
Such a factorization u is unique since / is a monomorphism. D
Lemma 2.6.9 An additive exact category ^ is finitely cocomplete.
Proof By 2.8.1, volume 1, and 2.6.4, this volume, it suffices to prove
the existence of coequalizers, which is equivalent to the existence of
cokernels (see 1.2.8). Given a morphism /: A >B, we factor it as
f = iop with i a monomorphism and p a regular epimorphism. Since
p is an epimorphism, the cokernel of i, which exists by 2.6.8, is also the
cokernel of /. D
Lemma 2.6.10 In an additive exact category, every epimorphism is a
cokernel.
Proof Let /: A >B be an epimorphism. Since the category is
regular, we can factor f as f = i op where i is a monomorphism and p is
a regular epimorphism (see 2.1.3). But the monomorphism i is a kernel
by 2.6.8; so it is a strong monomorphism by 4.3.6, volume 1; but i is
also an epimorphism, since / is. Finally i is an isomorphism (see 4.3.6,
volume 1) and / is a regular epimorphism. Thus / = Coker(ix, t') for
some pair ix, v: P ^A and therefore / = Coker {u — v); see 1.2.8. D
Theorem 2.6.11 The following conditions are equivalent:
A) ^ is an abelian category;
B) ^ is an additive exact category;
C) ^ is a non-empty, preadditive exact category
Proof By lemmas 2.6.2 to 2.6.10 and examples 2.4.1 and 2.5.5.d. D
Counterexample 2.6.12
Let us recall that an abelian group A (written additively) is torsion-free
when the property
na = a-\-...-\-a = 0 => a = 0
holds for each element a E A and each non-zero natural number n G N*.
Let us consider the category ^ of torsion-free abelian groups and group
homomorphisms between them. We shall prove that ^ is regular, but
not exact.
First of all observe that the product of two torsion-free groups is again
torsion-free as well as every subgroup of this product. This proves
already that the category of torsion-free groups has puUbacks which are
computed as in the category of abelian groups.

110 Regular categories
Next observe that given a homomorphism /: A >B between
torsion-free groups, the coequalizer of its kernel pair in the category of
abelian groups is given by the epimorphic part p of the mono-epi-
factorization f = i o p oi f (see 2.1.4). Since every subgroup of the
torsion-free group B is still torsion-free, this coequalizer is also the co-
equalizer in ^.
The previous observation shows also that if / is a strong epimorphism
in ^, i is both a monomorphism and a strong epimorphism in ^, thus an
isomorphism (see 4.3.6, volume 1). Therefore / is a strong epimorphism
in ^ precisely when it is a strong epimorphism in the category of abelian
groups.
All these observations, joined to the fact that the category of abelian
groups is regular (see 2.4.1), let us conclude that ^ is regular as well.
Since ^ is additive, the exactness of ^ would imply its abelianness
(see 2.6.11). In particular one would have a short exact sequence
0 >2Z ^ >Z 2 ^Q >o
where i is the canonical inclusion of the even integers in Z and q is the
cokernel of i in ^. But from ^^B) = 0 we deduce q{l) + q{l) = 0, thus
^A) = 0 since Q is torsion-free. This implies that g = 0, with therefore
the identity on Z as kernel (see 1.1.8). This contradicts the fact that i
should be the kernel of q (see 1.8.5).
2.7 An embedding theorem
The aim of this section is to prove that every small regular category ^
admits a full exact embedding in a Grothendieck topos: as a consequence,
to prove an exactness property in a regular category, it suffices to prove
it in every Grothendieck topos. And most often, in a topos, an exactness
property is proved just as in the category of sets, using the internal logic
of the topos (see chapter 6, volume 2).
In this chapter, we assume some familiarity with at least the
definition of a Grothendieck topos. Moreover we assume freely the axiom of
universes (see 1.1.4, volimie 1) so that every regular category can be
viewed as a small category with respect to some sufficiently big universe
of sets.
Proposition 2.7.1 Let ^ be a small regular category. For each regular
epimorpbism /: D ^►>C in ^, consider the subobject Rf C <<^(—, C) of

«/
2.7 An embedding theorem
-^ Rf
\rf
^<--*)fF;irf(-'^>
w
Diagram 2.10
u
111
■^ D
/
■^ C
those morphims of ^ vrhich. factor through f:
RfiX)^{g€^{X,C)\3he^{X,D) g = foh]
Those subobjects Rf constitute a localizing system in the sense of 3.2.1,
volume 3 (i.e. are puUback stable). We shall refer to it as the ^'localizing
system of regular epimorphisms".
Proof Consider a regular epimorphism /: D »C, an arbitrary mor-
phism u: A >C and the puUbacks of diagram 2.10, respectively in
Fun(^*,Set) and ^. By 2.1.3, w is a. regular epimorphism. Moreover
given g e ^(X, A), g factors through w iS uo g factors through /, just
by the definition of a puUback. With the notation of the statement, this
means
geRw^uogeRf^g e^{-,u)~^{Rf),
i.e. R^ =^{-,u)-^{Rf). n
Lemma 2.7.2 Let ^ be a small regular category. A contravariant
functor F: ^* >Set is a sheaf for the localizing system of regular epimor-
phisms iff it maps an exact sequence in ^ onto an equalizer diagram in
Set.
Proof We use the notation of 2.7.1. Fixing a regular epimorphism
/: D »C in ^, we write u^v for its kernel pair, yielding an exact
sequence m
u
to-
f
^>C.
->Set, a nat-
Given a presheaf (i.e. a contravariant functor) F: ^*
ural transformation a: Rf ^ F is completely determined by the single
element anif) G F{D). Indeed, given X G^ and g e Rf{X), g = f oh
for some h G ^(X, D) and therefore, by naturality of a,
axig) = axif o h) = {ax o Rf{h)){f) = {F{h) o az,)(/).

112 Regular categories
This element a = aoif) is such that F{u){a) = F{v){a)^ because
F{u){aD{f)) - {apoRf{u)){f) = ap{fou)
^ap{fov)^{apoRf{v))if) = F{v){aD{f)).
Conversely, given an element a e F{D) such that F{u){a) = F{v){a)
and arrows ^, h as before, the formula ax{g) = F{h){a) defines a natural
transformation a: Rf =^ F, just because, given x: Y >X,
{F{x) o ax){g) = {F{x) o F(/i))(a) = F{hox){a) = ay(/ ohox)
= (ay o ^(o:, C)) {foh) = (ay o <^{x, C)) {g).
In other words, the natural transformations a: Rf =^ F are exactly
determined by the elements a e F{D) such that F{u){a) = F{v){a).
By the Yoneda lemma, the natural transformations /3: ^(—,C) =^ F
correspond bijectively with the elements b G F{C)^ the correspondence
being given by (see 1.3.3, volume 1)
6 = /3c(lc), M9)-F{g){h).
In particular given h G ^(X, D) and g = f o h e i?/(X),
/?xE) = F(/o/i)F) = F(/i)(F(/)F)).
Obviously one has
F(«) (F(/)F)) = F(/ o «)F) = F{f o t;)F) = {F{v)) (F(/)F))
SO that a = F{f){b) e F{D) is the element which determines the
restriction a oi P to Rf,
In conclusion, every natural transformation a: Rf ^ F extends
uniquely as /3: ^(-, C) =^ F iff the correspondence
F{C) >F{D), b^F{f){b)
induces a bijection between F{C) and the elements o € F{D) such
that F{u){a) = F{v){a). This is equivalent to saying that the following
diagram is an equahzer in the category of sets:
F(f) ^(") ,
F{C) ^^^' >F{D) iFjP). D
F{v)
Theorem 2.7.3 Let ^ be a small regular category. The Yoneda emhed-
ding
Y',<€ >Sh(^,7i)

2.8 The calculus of relations 113
mapping C e ^ to Y{C) = ^(—,C) is a full exact embedding of^
in the topos of sheaves over ^ for the localizing system TZ of regular
epimorphisms.
Proof Let us first mention that every topos is indeed a regular category
(see 3.4.14, volume 3).
Given an exact sequence
/
to—i—»c
V
in ^, / = Coker {% v) so that ^(/, C) = Ker (^(ix, C), ^(t;, C)) by 2.9.5,
volume 1. Thus each ^(—,X) is indeed a sheaf so that the Yoneda
embedding, which is full and faithful, takes values in the topos of sheaves.
It remains to prove that this Yoneda embedding is exact. The Yoneda
embedding of ^ in its category of presheaves preserves finite limits (see
2.15.5, volume 1) and the topos of sheaves Sh(^, 72.) is stable under
finite limits in the topos of presheaves (see 3.4.3, volume 3). Therefore
the Yoneda embedding of the statement preserves finite limits.
By 2.5.7, volimie 1, it remains to prove that Y preserves regular
epimorphisms. If /: D »C is a regular epimorphism in the category
^ and a,/?: ^(—,C) >F are two morphisms in Sh(^,72.) such that
a o ^(—, /) = /J o <J^(—, /), a, /? correspond by the Yoneda lemma (see
1.3.3, volume 1) to elements a,b e F{C) such that F(/)(a) = F{f){b).
Since F is a sheaf, F{f) is a monomorphism and therefore a = b^ thus
a = /?. So ^(—, /) is an epimorphism, thus a regular epimorphism (see
3.4.13, volume 3). D
Metatheorem 2.7.4 To prove that a property involving just finite
limits and exact sequences holds in every regular category it sufGces to
prove it holds in every Grothendieck topos.
Proof The Yoneda embedding of 2.7.3 is full and faithful, thus it reflects
isomorphisms. Since Y preserves finite limits and exact sequences, it
reflects them as well (see 2.9.7, volume 1). D
2.8 The calculus of relations
In this section, we make constant use of the metatheorem 2.7.4 and
develop all our proofs in a Grothendieck topos via its internal logic (see
chapter 6, volume 3). It is an easy but lengthy exercise to write direct
proofe involving just the consideration of diagrams in the original regular

114
R
P2
-> s
pf
-> B
Regular categories
Diagram 2.11
p^ ov
^AxC
category. Let us mention once more that every topos is a regular category
(see 3.4.14, volume 3).
In this section, we shall use the notation R: A o >B to denote a
relation from A to B (see 2.8.1). A quick look at theorem 2.8.4 shows
that the analogy with the theory of distributors (see 7.8, volume 1)
justifies this analogy in the notation.
Finally let us recall that by a subobject, we mean an equivalence class
of monomorphisms, for the isomorphism relation (see 4.1.1, volume 1).
Definition 2.8.1 Let ^ be a finitely complete category By a relation
R: A o >B, we mean a suhobject RC Ax B. We write
pf: RCAxB^^A,
p^iRCAxB^^B
for the corresponding projections.
In a topos, pf and pf can thus be described by the formulas pf (a, b) =
^» P^(a? b) =b where a e A, b e B.
Definition 2.8.2 Let ^ be a finitely complete category. Given a relation
R: A o >B in ^, we define the opposite relation R^: B o >A as the
following composite:
RCBxA^AxB,
where the isomorphism is the canonical one. In other words, pf = pf
andpf =pf.
In a topos, RP can thus be described as
R^ = {{b,a) e B X A\{a,b) e R}.
Definition 2.8.3 Let ^ be a Bnitely complete and regular category.
Given two relations R: A—^^B and S: B-^o-^C in <g, their composite
SoR: A o >C is deRned as in diagram 2,11, where the left-hand square

2.8 The calculus of relations 115
is a puUback and the right-hand triangle is an image factorization (see
2,1,4),
In a topos, one thus has the following descriptions:
P = {{a,b,c) eAxBxC\{a,b)eR F,c) e S},
SoR={{a,c)eAxC\3beB {a,b) e R {b,c)eS},
Observe that every category ^ can be viewed as a "discrete" 2-
category with just identity 2-cells (see 7.1.4.d, volume 1). With that
in mind, we get the following theorem.
Theorem 2.8.4 Let ^ be a regular, well-powered and finitely complete
category. We get a 2-category Rel(^) by choosing as:
• objects = those of ^;
• arrows = the relations of^, with the composition defined in 2,8,3;
• 2-cells = the inclusions of relations, viewed as subobjects in ^,
Moreover there is an injective 2-functor p: ^ >Rel(^) with the
property that an arrow R: A o >B in Rel(^) has a right adjoint if and only
if it has the form p{f) for some f: A >B in ^; this adjoint turns out
to beR^,
Proof By our metatheorem 2.7.4, it suffices to write the proof in a
topos, using its internal logic.
Given an object A G ^, write A^ C Ax AioT its diagonal relation. If
R: A o >B is an arbitrary relation,
RoAA = {{a,b)eAxB\3a'eA {a,a') e Aa {a',b)eR}
= {(a, b)eAxB\ (a, b) e R} = R,
since (a, a') G A^ means a = a'\ in the same way Ab oR = R.To prove
the associativity of the composition law for relations, consider relations
R S T
-e >B e >C e >J9.
We have
To{SoR)
= {{a,d)eAxD\3ceC {a,c)eSoR {c,d)eT}
= {{a,d)eAxD\3ceC 3b e B {a,b) e R {b,c)eS {c,d)eT}
= {{a,d)eAxD\3beB {a,b) e R {b,d)eToS}
= {ToS)oR,

116 Regular categories
Thus the objects of ^, together with the relations, already constitute
a category Rel(^), with each Re\{^){A^ B) indeed a set since ^ is well-
powered.
But in fact each Rel(^)(i4, B) is a poset, thus a category, for the partial
ordering given by the inclusion of subobjects. Defining the Godement
product of 2-cells reduces to checking that given
RCSCAxB, TCUCBxC,
then ToRCUoS. Indeed
{a,c)eToR^3beB {a,b) e R F,c)eT
^3beB {a,b)eS {b,c)eU
=^ (a, c) G C/ o S.
The other conditions for having a 2-category are just obvious since they
require the equality of some 2-cells; and when a 2-cell exists between
two arrows, it is necessarily unique since each Rel(^)(i4, B) is a poset.
Thus Rel(^)(i4, B) is already a category.
The embedding p is defined by p{A) = A for every object A. Given
an actual morphism /: A >B of ^, its "graph" defined by
^^ Y,A >AxB
is a monomorphism, since its composite with the first projection is just
1a (see 1.7.2, volume 1). So Cf) is a relation from A to B, which we
choose as p(/). This relation p{f) can thus be described by the formula
p{f) = {{a,b)eAxB\b^f{a)}.
In particular
p{1a) = {{a,a') eAxA\a' = Ula)} = Aa
and if g: B >C is another morphism of ^
P(9) o Pif) = {{a,c)eAxC\BbeB (a,b) e p{f) F, c) e p{g)}
= {(a,c)eAxC|36€S b = f{a) g{b) = c]
= {(a,c)GAxC|36eB 6 =/(a) g{f{a))=c}
= {{a,c)eAxC\{9of){a) = c)
= Pi9°f)-
Thus p is indeed a functor.

2.8 The calculus of relations 117
By construction p is injective on the objects. Now if /, g: A ] E are
morphisms of ^ such that p{f) = p{g), then {^f) = {^^) and composing
with the second projection yields f = g. Thus the functor p is injective.
Next consider an arrow /: A >B. The relation p{f)^: B ^A is
right adjoint to the relation p{f): A >B in the 2-category Rel(^) of
relations (see 7.1.2, volume 1) precisely when
P(/)o/9(/)«CAb, AACpiffopif).
This is indeed the case since
pif)op{ff^{ib,b')eBxB\3aeA {b,a)ep{ff {a,b') e p{f)}
= {{b,b') e B X B\3a e A 6 =/(a) b'^ f{a)}
C{{b,b')eBxB\b^b'}
= Ab,
p{ffop{f) = {{a,a')eAxA\BbeB {a,b) e p{f) {b,a') e p{ff}
^{{a,a')eAxA\BbeB b = f{a) 6 =/(a')}
^{{a,a')eAxA\f{a) = f{a')}
2Aa.
Conversely consider two relations R: A o >B and S: B o >A such
that
RoSCAb, AaCSoR.
We must prove that R = p{f), for some /: A >B in ^. For this it
suffices to prove that
VaeA 3\beB {a,b)eR
(see 5.10.9, volume 3). And indeed, given o G A,
ae A^ {a,a) e Aa
^ {a,a) € SoR
^3beB {a,b)eR {b,a) e S
=^3beB (a, b) G R,
(a, b)eR (a, b') e R
=^ (a, b) eR (a, b') e R (o, a) G Aa
=> (a, b)GR (o, b') eR (o, a) € 5 o i?
^{a,b)eR {a,b')€R 3b" e B {a,b") € R {b",a)eS
=>36"eB {b",a)eS {a,b) e R {b",a)eS {a,b') € R

118
AxB
AxB
R
Diagram 2.12
^3b''eB {b'',b)eRoS {b'\b')eRoS
^3b''eB {b'\b)eAB {b'',b')eAB
^3b"eB b" = b b" = b'
^b = b'.
n
In fact, in 2.8.4, the arrows of the type p{f) and their adjoints suffice
to generate Rel(^).
Proposition 2.8.5 Let ^ be a regular, well-powered and finitely
complete category. Given a relation R: A o >B, consider the situation of
diagram 2.12 in the category ^. In the category Rel(^) of relations, R
can be written as the composite R = p{pB or) o p(p^ or)^.
Proof By our metatheorem 2.7.4, it suffices to write down the proof in
a topos, using its internal logic.
p{PB o r-) o p{pA o rf
= {(a, 6) eAxB\ 3{a\b') e R PA{a',b') = a psia',b') = b}
= {(a,6) eAxB\ 3{a\b') eR a' = a b' = b]
= {{a,b)eAxB\{a,b)eR]
= R. n
Another interesting property of the inclusion in 2.8.4 is a
characterization of monomorphisms and epimorphisms.
Proposition 2.8.6 Let ^ be a regular, well-powered and finitely
complete category. Given a morphism f: A >B in ^:
A) f is a monomorphism iff p{f)^ o p{f) = A^;
B) f is a regular epimorphism iff p{f) o p{f)^ = Ab-

2.8 The calculus of relations 119
Proof By our metatheorem 2.7.4, it suffices to write down the proof
in a topos, using its internal logic. We recall that in a topos, every
epimorphism is regular (see 3.4.13, volume 3).
We know already that given /: A >B,
Piff o p{f) = {(a, a')eAx A\ f{a) = /(a')},
p{f)op{ff = {{b,b')eBxB\3aeA b = f{a) b'= f{a)}
= {{b,b)eBxB\3aeA b = f{a)}
(see proof of 2.8.4). It follows immediately from these formulas that
p{ff o p{f) = Aa iff Va, a'eA f{a) = f{a') ^a = a'
iff / is a monomorphism
p(/)op(/H = ABiffV6GB 3a e A b = f{a)
iff / is an epimorphism
(see 5.10.2, volume 3). D
Let us conclude this section with the so-called "modularity laws" for
relations. They are identities satisfied in the category Rel(^) and
particularly useful when computing in this category.
Proposition 2.8.7 Let ^ be a regular, well-powered, finitely complete
category Consider three relations
R: A e >B, S: B e >C, T: A e >C,
The following identities hold:
{S o R)nT C S o {Rn{S^ oT)),
{SoR)nTC{Sn{To R^)) o R,
where fl denotes the intersection as subobjects in ^.
Proof Once more we use our metatheorem 2.7.4 and the internal logic
of toposes. We prove the first relation; the second one is analogous.
{SoR)nT
= {{a,c)eAxC\{a,c)eSoR {a,c)eT}
= {{a,c)eAxC\3beB {a,b) e R {b,c)eS {a,c)eT}
= {{a,c)eAxC\3beB {a,b) e R
{a,c)eT {c,b)eS^ {b,c)eS}
C{{a,c)eAxC\3beB {a,b)eR
Cc' {a,c')GT {c\b)GS^) {b,c)GS}
= {{a,c)eAxC\3beB {a,b) e R {a,b)eS^oT {b,c)eS}

120 Regular categories
= {{a,c)£AxC\3b£B {a,b) £ Rn{S^ oT) {b,c)GS}
= So{Rn{S^oT)). D
2.9 Exercises
2.9.1 In the category of abeUan groups, prove that pulling back along a
morphism does not respect coequalizers. [Hint: consider the two
canonical injections 5i, 52: A > A © A of a biproduct and pull their coequal-
izer back along a zero morphism.]
2.9.2 Let us call "universal" an epimorphism /: A >B whose pull-
back along every morphism g: C >B exists and is again an
epimorphism. Show that in definition 2.1.1, the last axiom cannot equivalently
be replaced by "every regular epimorphism is universal" [Hint: observe
that in the category of topological spaces, every epimorphism is
universal]. Nevertheless, show that making this replacement, every morphism
still factors as a regular epimorphism followed by a monomorphism.
2.9.3 Prove that the category of sets is both exact and coexact.
2.9.4 Prove that the category of compact Hausdorff spaces is both exact
and coexact.
2.9.5 Describe, in terms of finite limits, what it means for a relation
ri,r2: ^ > A on an object A of a category to be a partial order.
2.9.6 In a category with binary products, prove that the two projections
PijP2- ^ X A ^A constitute an equivalence relation on A,
corresponding to the identity subobject A x A===A x A.
2.9.7 Prove that every poset is an exact category.
2.9.8 Show that the category of compact Hausdorff 0-dimensional spaces
is regular but not exact.
2.9.9 Let ^ be a finitely complete regular category. An object C € ^ has
global support when the morphism C »1 is a regular epimorphism.
Prove that the full subcategory ^ of objects with global support is again
regular, but not in general finitely complete. When ^ is exact, ^ is exact
as well.
2.9.10 Let ^ be a regular category. Prove that the following conditions
are equivalent:
A) if Jf?,S are two equivalence relations on an object A, then Ro S =
SoR;
B) if R: A'-^e-^A is a relation, then R = RoR^ oR;

2.9 Exercises 121
C) every reflexive relation RC Ax Ais symmetric;
D) every reflexive relation RC Ax A is an equivalence relation;
E) if i2,S are two equivalence relations on an object A, then Ro S is
also an equivalence relation on A,
An exact category which satisfies these equivalent conditions is called a
Mal'cev category.
2.9.11 Let ^ be a finitely complete category with strong-epi-mono
factorizations. Prove that ^ is regular iff the composition of relations is
associative. [Hint: given an epimorphism /: A »C and an arbitrary
morphism g: B >C, consider the composite p(/) o p(/)^ o p{g).]

3
Algebraic theories
In this chapter, we investigate a general approach to those structures like
groups, rings, modules, lattices, boolean algebras... characterized by the
existence of one or several operations which are defined everywhere and
satisfy axioms expressed by equalities.
Let us immediately underline the fact that structures like fields and
categories do not fall under the scope of this chapter, even if they have an
obvious algebraic nature. Indeed, the theory of fields admits an operation
(the inverse for the multiplication) which is not defined everywhere and,
clearly, the composition law in the case of a category is just defined for
some pairs of arrows (see exercises 3.13.1 and 3.13.2), not for all of them.
3.1 The theory of groups revisited
A group can be defined as a set G provided with a binary operation
-fiGxG >G
satisfying the axioms
yx,y,z gG (x -f 2/) -f z = X -f B/ -f z),
30 gG Vx€G 3y GG x-hO = x = 0-hx , x-\-y = 0 = y-\-x.
The existence of the unit or the opposite can in fact be presented as
an axiom, not as a property, yielding the equivalent definition that a
group is a set G provided with
• a binary operation +: G x G >G,
• a l-ary operation —: G >G,
• a constant 0 € G,
122

3.1 The theory of groups revisited 123
satisfying the axioms
\/x,y,z £G {x -\- y) -\- z = X -\- {y -\- z),
Vx € G X + (-x) = 0 = (-x) + X.
One should observe that these axioms are now presented in a very
elementary form: just equalities between algebraic composites, without
any existential quantifier, implication symbol, conjunction, disjunction
or negation.
As usual, we shall write G^ for the product of n copies of G, n €
N; in particular G^ is the singleton, as observed in 2.3.2.a, volume 1.
Observe also that giving a constant 0 € G is equivalent to giving a
-ary operation" 0: G^ >G, so that finally the theory of groups has
three operations,
• a binary operation -f: G^ >G,
• a 1-ary operation —: G^ >G,
• a 0-ary operation 0: G^ >G,
and the axioms can be expressed by the commutativity of the various
pieces of diagram 3.1.
The previous presentation is somewhat misleading since it does not
describe the "theory of groups", but the "theory of a group G"; indeed,
our description uses explicitly a "generic" set G to describe the group
structure. It is easy to overcome this difficulty. The theory of groups
can be described by giving a denumerable set of variables x, y, z,... as
well as three formal symbols +, —, 0 and the equality. The terms of the
theory of groups are then defined inductively by the following rules:
• every variable is a term;
• if 5, t are terms, 5 -f t is a term;
• if t is a term, —t is a term;
• 0 is a term.
The axioms of the theory of groups are expressed by equalities between
terms: if x, y, z are variables,
X + B/ -f z) = (x + 2/) + z,
X + 0 =x = 0 + X,
X + (—x) =0 = (—x) + X.

124
Algebraic theories
C?)
+
+
-> G^
G^ X G"^
IgxO
G^
^- G' —^
■^G" X G^
+
-^ G^ ^
+
OxIg
G2
1 -^^ G2 ^-^i^ r;i
+
GO
Diagram 3.1
G«
A model of this formal theory of groups is now a set theoretical
interpretation. More precisely, we fix a set G and interpret the three symbols
+, —, 0 by choosing
• a mapping -j-: G x G >G,
• a mapping —: G >G,
• an element 6 G G,
where we have written a-i-6 for -i-(a, b). The terms are then interpreted
inductively:
• a variable can be interpreted as any element of G;
• if the terms s,t are already interpreted as elements |5|, \t\ G G, the
term s -f-1 is interpreted as the element |5| -j- |t|;
• if the term t is already interpreted as an element \t\ G G, the term
—t is interpreted as — |t|;
• the term 0 is interpreted as the element 6 G G.

3.2 A glance at universal algebra 125
The data (G, -f, —, 0) constitute a model of the theory of groups when,
for each possible interpretation of the variables, the three equalities
N+(M + N) = (N + l2/l)+N,
|x|+6 = |x| = 6+|x|,
|x|-f(-|x|) =6= (-|x|)+|x|
hold between elements of G, for all variables x, y, z.
3.2 A glance at universal algebra
This section is intended to explain the precise link between classical
universal algebra and categorical universal algebra. The reader just
interested in the latter can go directly to section 3.3.
What has been done in 3.1 could be repeated for the theory of rings
with unit: it would be necessary to add a second constant 1 and a second
binary operation x. And clearly one can easily imagine theories where
one has ternary or even n-ary operations, for n € N. This leads to the
following general definition.
Definition 3.2.1 A presentation of an algebraic theory T is a theory
with equality, specified by choosing, besides a denumerable set of
variables, a set On of ^'n-ary operations" for each integer n € N, together
with a set of axioms, subject to the following requirements. The terms
of the theory are defined inductively:
• each variable is a term;
• ifa£ On and ti,..., t^ are terms, then a(ti,..., t^) is a term.
An axiom is an equality between two terms.
Observe that the second condition defining the terms, in the case
n = 0, becomes exactly
• ii a G Ooi then a is a term.
The 0-ary operations are also called constants; thus
• every constant is a term.
Definition 3.2.2 Let T be a presentation of an algebraic theory in the
sense of 3,2,1, By a model ofT we mean the choice of
• a set My
• for aJi n G N, for all a e On, a mapping \a\: M^ >M,

126 Algebraic theories
in such a way that the axioms of T are realized by this interpretation.
More precisely
• a variable is interpreted as any element of M,
• if a € On and the terms ti,..., tn are already interpreted as elements
\ti\i" ' A^n\ ^ M then a(ti,..., t^) is interpreted as the element
\a\{\ti\,...,\tn\)eM,
and an axiom is satisfied in M when, for every possible interpretation
of the variables, both sides of the equality have the same interpretation
in M.
Definition 3.2.3 Let T be a presentation of an algebraic theory in the
sense of 3.2.1. If L^M are models ofT, a T-homomorphism f: L >M
is a mapping f: L >M such that for every operation a € On and
every elements xi,..., x^ € L
/(|a|(a:i,...,Xn)) = |a| (/(xi),... ,/(xn)).
It is a completely obvious observation that
Proposition 3.2.4 Let T be a presentation of an algebraic theory in
the sense of 3.2.1. The models ofT and their homomorphisms, together
with the usual composition of mappings, constitute a category.
What we have just described is the precise object of universal algebra.
Now we would like to give an equivalent and very elegant categorical
presentation of these notions. To achieve this, we need a series of elementary
observations.
Lemma 3.2.5 Let T be a presentation of an algebraic theory. There
exists a smallest equivalence relation R on the set of terms such that:
A) if the axiom s = t holds, then the pair (s^t) is in R;
B) if the terms 5, t are written using the variables Xi,..., x^, the pair
E, t) is in R and ti,..., t^ are terms, then the pair {s\ t') is in R,
where s\ t' are obtained from s^t by replacing Xibyti,i = l,...,n;
C) if a E On and the pairs {si, ti) are in i?, (i = 1,..., n), then the pair
{a{si,..., Sn), a{ti,..., tn)) is in R.
Proof Just construct R inductively from the pairs (s^t) given by the
axioms, using conditions B), C) and the requirements for an equivalence
relation. D
Lemma 3.2.6 Let T be a presentation of an algebraic theory with set
{xi,..., Xn,...} of variables. Let Tn be the set of terms involving only
the variables xi,..., Xn- Let Fn be the quotient ofTn by the (restriction

3.2 A glance at universal algebra 127
of the) equivalence relation R of lemma 3.2.5. The set Fn is naturally
provided with the structure of a T-model.
Proof If a € Om and ti,..., t^ € Tn, then clearly a(ti,..., tm) € Tn.
By the last condition defining i?, this construction is compatible with
the equivalence relation, yielding an interpretation
\a\:{Fnr >Fn.
This interpretation satisfies all the axioms of T whatever the
interpretations chosen for the variables are, just by the first two conditions defining
the relation R. D
Lemma 3.2.7 Let T be a presentation of an algebraic theory. In the
category Modr of T-models, Fn is the n-th copower of Fi.
Proof We use the notation of 3.2.6. For each variable Xi and index
1 < i < n, consider all the equivalence classes of all the terms involving
just the variable x^; this yields a T-model F^^^ isomorphic to Fi, together
with an inclusion
Si:F« >Fn
which is obviously a T-homomorphism.
Now consider a T-model M together with a family of T-homomor-
phisms fi'. F^^^ >M, i = l,...,n. Interpret the variable x^, i =
1,..., n, in M as /i ([x^]), where [xi] is the equivalence class of the term
Xi in Fn. Define a mapping Tn >M by mapping the term t to its
interpretation in M corresponding to the previous interpretation of the
variables. Since M is a T-model, this construction is compatible with the
axioms of T and since each fi is a T-homomorphism, this
compatibility extends to the whole relation i?, yielding finally a T-homomorphism
/: Fn >M. By construction, f o Si = fi. On the other hand this
relation yields f{[xi]) = fi{[xi])^ from which we get the uniqueness of /.
D
Lemma 3.2.8 Let T be a presentation of an algebraic theory. The
model Fn is the free model on n generators.
Proof We consider the "underlying set functor"
U: Modr >Set
mapping a model M to the underlying set M. The statement means that
Fn, together with the mapping
{l,...,n} >Fn, i^[xi],

128 Algebraic theories
is the reflection of {1,..., n} along U.
Giving a mapping {1,..., n} >U{M) in the category of sets is just
giving n elements ai,..., a^ € M. Those elements are then chosen as
interpretations of xi,...,Xn in M. This yields a mapping Tn >M
mapping a term t to its interpretation in M corresponding to the
previous interpretation of the variables. Since M is a model, this construction
is compatible with the equivalence relation R of 3.2.5, yielding the
required morphism Fn > M. D
Here now is the key to a categorical approach to universal algebra.
Proposition 3.2.9 Let T be a presentation of an algebraic theory
With the notations of 3,2,7, write ^ for the full subcategory of Modr
generated by the free models Fn on finitely many generators. The dual
category ^* has finite products and Modr is equivalent to the category
of finite product-preserving functors from ^* to the category of sets,
and natural transformations between them.
Proof For clarity, we work with the category ^ which has finite
coproducts (see 3.2.7) and consider the contravariant functors ^ >Set
transforming finite coproducts into finite products. By 3.2.8, a morphism
Fn >Fm in ^ is just the choice of n elements of Fm-
Given a contravariant functor G: ^ >Set transforming finite co-
products into finite products, let us construct a T-model by putting
M = G{Fi), Every n-ary operation a yields an element [a(xi,..., Xn)]
in Fn, thus a T-homomorphism a: Fi >Fn. This induces a mapping
in Set G{a): G{Fn) >G{Fi), By assumption on G, G{Fn) ^ (G(Fi))''
so that G{a) is in fact a mapping |a|: M'^ >M, yielding the
interpretation of a in M.
Now suppose the variables xi,..., x^ are interpreted in the model M
as elements ai,..., an- The interpretation of the term a(xi,..., Xn) in
M is just |a| (ai,... ,an). By induction, if t is a term with variables
^ij • • • j^nj then [t] 6 Fn corresponds to a morphism t: Fi >Fn and
the interpretation of [t] is G(?)(ai,..., a^) € M, In particular if s, t are
terms with variables among {xi,..., Xn} and the axiom s = t holds, the
relation [s] = [t] holds in Fn by definition of the relation R (see 3.2.5);
therefore I = f and finally G{'s){ai,..., a^) = G(?)(ai,..., a^), proving
that s and t have the same interpretation in M, Finally, M is indeed a
T-model.
Conversely, let us start with a T-model M, We define G: ^ >Set
on the objects by G{Fn) = M^. A morphism /: Fn >Fm corresponds
to the choice of n elements [ti],..., [tn] G Fm where the terms ti,..., tn

3,2 A glance at universal algebra 129
have variables xi,...,Xm- In the category of sets, we must construct
a mapping G{f): M'^ >M'^. Given an m-tuple (ai,... ,am) € Af^,
we interpret each variable x^ (z = 1,..., m) as a^ and map the m-tuple
(ai,..., am) to the n-tuple (| ti|,..., | t^l) in M'^ corresponding to that
interpretation of the variables. Clearly G is a functor. Moreover, with the
notation of 3.2.7, Si: Fi >Fm corresponds to the element [xi] € Fm\
the corresponding mapping G{si)'. M'^ >M maps therefore the m-
tuple (ai,..., am) to the interpretation of the term x^, i.e. to a^. Thus
G{si) is just the z-th projection and G transforms finite coproducts into
finite products.
The argument is now easily extended to the morphisms. Indeed given
two functors G, H: ^ >Set transforming finite coproducts into finite
products, a natural transformation /x: G => if is a family of mappings
/jin'' G{Fn) >H{Fn) Satisfying in particular the equality
H{si) ofi^ = fiio G{si) (z = 1,..., n).
But since H{si) and G{si) are just the z-th projections, this means that
fin is just the n-th power of /xi. For every n-ary operation a, we can then
consider the corresponding morphism a: Fi >Fn and the relation
H{a) o (/xi)- = H{a) o /x, = /xi o G{a)
proves that /xi: G{Fi) >H{Fi) is a T-homomorphism.
Conversely, given the T-homomorphism /xi: G{Fi) >H{Fi)^ we
define fin'- G{Fn) >H{Fn) as the n-th power of /xi, yielding not only the
relations
H{Si) O /Xn = H{Si) O (/Xi)'' = /Xi O G{Si)
for every variable x^, but also
H{a) o nn = H{a) o (n^)^ = n, o G{a)
for every n-ary operation a, since /xi is a T-homomorphism. By
induction, this extends to
if(?)o/Xn = /xioG(?)
for every term t € T^. This suffices to prove the naturality of /x since
every /: Fn >Fm is completely determined by its composites with the
canonical morphisms Sii Fi >Fn of the coproduct. D
To conclude this section, let us observe that given a presentation of an
algebraic theory T, we can define a new presentation T' of an algebraic
theory by choosing as n-ary operations all the terms of T^ (notation of

130 Algebraic theories
3.2.6) and as axioms all the equations s = t where the pair E, t) belongs
to the relation R described in 3.2.5. Obviously, every T'-model is a T-
model. Conversely, if M is a T-model and t € Tn is a T-term, one gets
a corresponding interpretation of the T'-operation t
\t\: W >M
in the usual way: choosing (ai,..., a^) € M'^ we interpret Xi as a^ and
map (ai,..., a^) to the corresponding interpretation of t in the T-model
M. Since M is a T-model, one has |5| = \t\ for every pair {s^t) € i2,
proving that T and T' have the same models. In some sense, T' is a
"saturation of T" for all the possible operations and axioms.
But one can get an even more canonical presentation T" of an
algebraic theory, again having the same models as T and T', by choosing
now the elements of Fn as n-ary operations. For every pair E, t) € i2, the
axiom s = t oiT' induces a corresponding axiom [s] = \t\ for T"..., but
this axiom does not say anything relevant since it has the form w = w,
for some w € F^. In fact, for every term t £Tn constructed inductively
from the variables xi,..., x^ and the operations of T, one can consider
both the T''-term [t] £ Fn and the T''-term {t) 6 Fn obtained by
replacing, in the inductive construction of t, each m-ary T-operation a
by the corresponding T''-term [a(xi,..., Xn)]; one must choose all the
equations [t] = (t) as axioms for T''. We leave the details to the reader
who will observe that this last remark is essentially the content of 3.3.4.
3.3 A categorical approach to universal algebra
With 3.2.9 in mind, we define:
DeiSnition 3.3.1 By an algebraic theory T we mean a category T with
a denumerable set {T^, T\ ..., T"",...} of distinct objects, each object
T^ being the n-th power of the object T^. A model ofT is a functor
F: T >Set which preserves finite products, A homomorphism ofT-
models is a natural transformation.
We shall write Modr for the category of T-models.
Lemma 3.3.2 Let T bean algebraic theory. If a: F =^ G isa morphism
in Modx? ^he square in diagram 3.2 commutes, where the isomorphisms
are the canonical ones.

3.3 A categorical approach to universal algebra 131
Diagram 3.2
Proof For every projection pi: T^ >T^ of the n-th power (T^)'^, one
has
G{pi) o arrr- = ari o F{pi) = G{pi) o (ar^)'^
since the morphisms G{pi) and F{pi) are just the i-th projections of the
powers (G(Ti))'', {F{T^))''. As a consequence ar^ ^ {ari)''. D
Proposition 3.3.3 Let T be an algebraic theory. Consider the functor
U: Modr >Set
of evaluation at T^. Then:
A) U is representable by T{T^, -);
B) U is faithful;
C) U reflects isomorphisms;
D) each finite set with n elements (n € N) admits T{T'^^ —) as a
reflection along U;
E) T(T^^ —) is a strong generator for Modr-
Proof Each representable functor preserves finite products by 2.9.4,
volume 1; therefore each functor T{T'^, —) is a T-model. By the Yoneda
lemma, given a T-model F,
Nat(T(T\ -), F) ^ F{T^) ^ U{F),
proving that U is represented by T(T^^ —). In the same way
Nat(T(T^,-),F) ^F(T^)
^ {F(T^)y
-Set({l,...,n},F(ri))
^Set({l,...,n},t/(F)),
proving that T{T'^^ —) is the reflection of the set {1,..., n} along U; see
3.1, volume 1.

132 Algebraic theories
Now consider a morphism of T-models a: F =^ G such that a^i is
bijective, i.e. an isomorphism in Set (see 1.9.6, volume 1). By 3.3.2,
a^n = (o^tO'^5 ^hus each ar^ is bijective as well. Therefore when U{a) =
a^i is an isomorphism, so is the natural transformation a and U reflects
isomorphisms. In particular T{T^^—) is a strong generator (see 4.5.13,
volume 1).
Observe also that given two morphisms of T-models a^/3: F => G,
the relations qt^ = (a^i)'^, /3t^ = (/JtO'^ show at the same time that
Ua = UC implies a = /?, proving the faithfulness of U. D
Given an algebraic theory T, we easily get a presentation TJ of this
algebraic theory by choosing the elements of T{T'^^T^) as n-ary
operations for 71. Next, given a 71-term t with variables among xi,... ,Xn,
one defines inductively the n-ary operation t € T{T'^^T^) associated
with t:
• the i-th projection T^ >T^ is associated with x^;
• if the operations /?i,..., /3m are associated with terms 5i,..., 5m and
a is an m-ary operation, the composite ao (/3i,..., /3^) is associated
with the term a{si^..., Sm)-
One chooses then t = t(xi, ..., Xn) as an axiom for 71.
A warning is necessary here. Clearly, if (T'^;pi,... ,pn) is the n-th
power of T^ and a: T^ >T'^ is an isomorphism, (T'^,pioa,... ^pnoa)
is also an n-th power of T^, so that the previous description refers to
one specified choice of projections. In particular, when n = 1, every
isomorphism a: T^ >T^ is such that (T\a) is the -th" power of
T^. But among all possible choices, there is now a canonical one, namely
a = Iq-i , the identity arrow on T^ in the category T. It is convenient
to assume that in the description of 71, this canonical choice has been
made. In other words, we require the axiom
1^1 (x) = X
for the presentation 71 associated with the algebraic theory T.
Proposition 3.3.4 Let T be an algebraic theory and consider the
corresponding presentation 71 of this algebraic theory The categories of
models for T and 71 are equivaJent and, via this equivalence, the
functor
U: Modr >Set
of 3.3.3 maps a Tl-modeJ to its underlying set.

3.3 A categorical approach to universal algebra 133
Proof If F: T >Set is a T-model, the set F{T^) provided with the
operations
F{a): F(Ti)^ ^ F(T^) >F{T^)
for each a € T{T'^^T^) is obviously a 7i-model; indeed, the satisfaction
of the axioms is just the functoriality of F, Moreover it follows
immediately from 3.3.2 that giving a morphism (p: F =^ G between T-models
is equivalent to giving a morphism (pxi: F{T^) >G{T^) between the
corresponding TJ-models.
It remains to prove that every 71-model M is isomorphic to the model
F{T^) arising from a T-model F. Just define F on the objects by
F(T^) = M^. Next if a: T^ >T^ is a morphism in T, a is also a
n-ary operation of 71 and one defines
F{a): F{T'') = W >M = F(T^)
to be the realization of the 7i-operation a in the model M. More
generally, given a: T^ >T'^ one defines
F{a): FiT"") ^ W >M^ ^ F(T^)
to be the m-tuple (F(pioa),..., F{pmoa)). The validity of the 71-axiom
in M indicates that the functor F preserves a priori the composites of the
form T'^ >T'^ >T^, but this implies immediately the preservation
of all composites, just by definition of F{a) for a: T^ >T'^. The
preservation of the identity on T^ follows from the axiom I7-1 (x) = x of
71; applying 3.3.2, this implies the preservation of the identity on each
T^. D
Examples 3.3.5
3.3.5.a The considerations of 3.1 have described a presentation of the
theory of groups, from which we derive a corresponding algebraic theory,
applying 3.2.9; analogous observations hold for the theories of monoids,
abelian groups, rings with or without a unit, commutative or not, and
so on.
3.3.5.b The most elementary presentation of an algebraic theory is
that without any operation and any axiom; a model of it is therefore
just a set. Applying 3.2.8 and 3.2.9, we conclude that the corresponding
algebraic theory is the dual of the category of finite sets.
3.3.5.c A poset structure on a set X is given not by operations on
X, but by a relation on X. In fact, the theory of posets is not algebraic
(see exercise 3.13.3). Nevertheless if one requires the poset (X, <) to be

134 Algebraic theories
a A-semi-lattice, its structure can now equivalently be defined by the
binary "meet" operation
A:X xX >X
which satisfies the axioms
X A {y A z) = {x A y) A z^
X Ay = y Ax^
X Ax = X.
Indeed, the poset structure is determined by the meet operation via the
relation
X <y iff X Ay = X.
Conversely, given a set X together with a meet operation satisfying the
three properties we have indicated, the relation x<2/iffxA2/ = xisa
poset structure on X for which x Ay is the infimum of the pair (x,2/).
Indeed, the relation is reflexive since x A x = x; it is transitive because
X Ay = X and y A z = y imply
xAz = {xAy)Az = xA{yAz)=xAy = X',
it is antisymmetric since x Ay = x and y Ax = y imply
x = xAy = yAx = y.
To observe that x A y is the infimum of the pair (x, y) for the poset
structure, notice that
xA(xA2/) = (xAx)A2/ = xA2/
implies x Ay < x and in the same way x A y < y, by symmetry of A;
moreover if z < x and z <y^
z = z A z = {z A x) A {z A y) = z A {x A y)^
proving that z <x Ay. This proves that the theory of A-semi-lattices is
algebraic. A top element can easily be introduced as a constant 1 (i.e. a
O-ary operation) together with the axiom
X A 1 = X.
In the same way a bottom element is required by introducing a constant
0 together with the axiom
0 A X = 0.

3.3 A categorical approach to universal algebra 135
3.3.5.d A lattice is a set provided with the structure of both a A-
semi-lattice and a V-semi-lattice, those two structures arising from the
same poset structure on X. The theory of lattices is algebraic,
admitting a presentation given by two binary operations A, V which are both
associative, commutative and idempotent (see example 3.3.5.c); these
operations are connected by the axioms
X A (x V 2/) = X,
{xAy)\/y = y.
We must check that these two additional axioms suffice to prove that
the two following poset structures coincide:
one has
X <y
x^y
=>
=>
x<2/iffxA2/ = x,
X ■<y \S. xM y = y.
y = {xAy)\/y = x\/y
x = xA{x\/y) = xAy
=>
=>
x^y,
x<y.
3.3.5.e Prom the two previous examples, one easily defines the theory
of distributive lattices, by adding the axioms
X A B/ V z) = (x A 2/) V (x A z),
X V B/ A z) = (x V 2/) A (x V z),
or the theory of boolean algebras, by adding a 1-ary operation (—)*
satisfying the axioms
xVx* = 1, xAx* =0.
All these theories are therefore algebraic.
3.3.5.f Let us now consider the theory of complete \/-lattices, i.e. of
those posets (X, <) where every subset has a supremum; the morphisms
are the mappings preserving those suprema, thus also the poset
structure. This is typically a non-finitary theory: the operation of "taking the
supremum" applies not to a finite set of elements, but to an arbitrary
set of elements. And indeed the theory of complete V-l^^^ices is not
algebraic in the sense of 3.3.1 (see exercise 3.13.4).
3.3.5.g If J? is a ring, an J?-module is an abelian group M together
with a scalar multiplication
R X M >M.

136 Algebraic theories
Such a scalar multiplication does not have the required form of an
operation M'^ >M. Nevertheless the theory of i?-modules is algebraic.
It is obtained by adding to the theory of abelian groups one 1-ary
operation f for every element r £ R, The interpretation of f in the case
of an i?-module M is just the scalar multipUcation by r. The additional
axioms in the case of left i?-modules are thus
f{x-\-y) =f{x)-\-f{y),
f{s{x)) = {fs){x),
(rT~s) (x) = f{x) -f s{x),
and when R has a unit
l(x) = X.
3.3.5.h If G is a group (written multiplicatively), a G-set is a set X
provided with an action of G,
GxX >X, {g,x)y-^gx,
satisfying the two axioms
{99')^ = 9{9'^)^
Ix = X,
For a fixed group G, the theory of G-sets is algebraic and admits a
presentation given by a 1-ary operation g for every element g G G^
together with the axioms
^{x) = 9(g'{x)),
l(x) = X.
Observe that the empty set is always a model of this theory.
3.3.5.i Consider the presentation of an algebraic theory having just
one single constant and no axioms at all. The models are the pointed
sets, i.e. the pairs (X, x) where X is a set and x € X is an element of
X; a morphism /: (X, x) >{Y, y) is just a mapping /: X >Y such
that f{x) = y,
3.3.5.J Consider the category T whose objects are the spaces W^,
(n € N), and whose morphisms are the ^^ functions between them;
this is obviously an algebraic theory. The models of this theory are
called the ^°^ algebras. Given a ^°° differentiate manifold V, the set
*'°°(V,R) of ^°° real-valued functions on V can be canonically provided

3,4 Limits and colimits in algebraic categories 137
with the structure of a ^^ algebra: given ^^ mappings a: W^ >R
and /i,..., fn- V >]R, one defines the composite a(/i,..., fn) just as
/i
V " '"" ^ yW ^ >]R.
3.4 Limits and colimits in algebraic categories
Proposition 3.4.1 Let T be an algebraic theory. The category Modr is
complete and limits are computed pointwise. In particular, the forgetful
functor U: Mod^? >Set preserves and reflects limits.
Proof By 2.15.1, volume 1, limits in Fun(T, Set) are computed point-
wise; by 2.12.1, volume 1, the limit of a diagram constituted of finite
product-preserving functors is again a functor-preserving finite products.
This proves that Modr is complete and that limits in Modr are
computed pointwise. In particular evaluating at T^ preserves limits, showing
that U preserves limits (see 3.3.3). Since U reflects isomorphisms, U
reflects limits as well (see 3.3.3, this volume, and 2.9.7, volume 1). D
Proposition 3.4.2 Let T be an algebraic theory. The category Modr
has Altered colimits and these are computed pointwise. In particular, the
forgetful functor U: Mod^? >Set preserves and reflects Altered
colimits.
Proof By 2.15.1, volume 1, colimits in Fun(T, Set) are computed point-
wise; by 2.13.4, volume 1, the colimit of a filtered diagram constituted
of finite product-preserving functors is again a functor preserving
finite products. This proves that Modr admits filtered colimits which are
computed pointwise. In particular evaluating at T^ preserves filtered co-
limits, showing that U preserves filtered coUmits (see 3.3.3). Since U
reflects isomorphisms, U reflects filtered colimits as well (see 3.3.3, this
volume, and 2.9.7, volume 1). D
Corollary 3.4.3 Let T be an algebraic theory. In Modr, &nite limits
commute with filtered colimits.
Proof The property holds in Set (see 2.13.4, volume 1) and both finite
limits and filtered colimits are computed pointwise in Modr see 3.4.1,
3.4.2. n

138 Algebraic theories
Corollary 3.4.4 Let T be an algebraic theory. The forgetful functor
U: Mod^z >Set preserves and reflects monomorphisms.
Proof Now U preserves and reflects isomorphisms (see 3.3.3) and kernel
pairs (see 3.4.1), thus also monomorphisms (see 2.5.6, volume 1). D
Theorem 3.4.5 Let T be an algebraic theory. The category Modr is
reflective in the category Fun(T, Set) and therefore it is both complete
and cocomplete.
Proof Fun(T, Set) is both complete and cocomplete (see 2.15.4,
volume 1), so that the last assertion follows from 3.5.3,4, volume 1. It
remains to prove that Mod^ is reflective in Fun(T, Set). We know
already that Modr is complete and the inclusion in Fun(T, Set) preserves
limits (see 3.4.1); by the adjoint functor theorem (see 3.3.3, volume 1),
it remains to check the solution set condition.
Consider two functors F, G: TZIZ^Set and a natural transformation
if: F =^ G; suppose G preserves finite products. Consider the set X
of all elements of G{T^) of the form (pT'^{x) for some x € F{T^); the
cardinality of X is bounded by that of F{T^). Consider now the set
Y of all elements of G{T^) of the form a(xi,... ,Xn), for some
integer n, n-ary operation a and elements x^ € X; the cardinality of Y
is bounded by that of LIn€N^(^'''^^) ^ ^""^ ^^us by the cardinality
of LIn€N^(^'''^^) X FiT^)"". By construction, Y is stable for all the
T-operations, thus putting
if(T^) = y^ C G(Ti)^ ^ G(T^)
defines a subfunctor H C G which, by construction, preserves finite
products. Observe that (p factors through H since given x € F{T'^), one
has
G{pi){^TAx)) = cpri{F{pi){x)) € Y
and thus cpTn (x) € Y'^. Therefore one gets a solution set for F by
choosing those finite product-preserving functors H: T >Set such that
H{T^) is a subset of some fixed set Y with cardinality less than the
cardinality of lIn€N^(^'''^^) x ^(r^)^. Once H{T^) is fixed, there is
indeed (up to a canonical choice of products) just a set of possibilities for
constructing a product-preserving functor H since, for every morphism
f3: jT* >T^, there is just a set of possible candidates for if (/3), i.e.
the set of mappings if (T^)^ >if (T^)^. D

3.5 The exactness properties of algebraic categories 139
3.5 The exactness properties of algebraic categories
The main purpose of this section is to prove that algebraic categories are
regular and even exact in the sense of chapter 2. Using proposition 3.3.4,
we find it more convenient to work with the presentation 71 associated
with the algebraic category T.
Lemma 3.5.1 Let T be an algebraic theory and 71 the corresponding
presentation of this algebraic theory. In Modr, an equivalence relation
on an object M is a sub-Ti-model R C M x M which is an
equivalence relation in the category of sets. Such an R is generally called a
^^congruence^^ on M.
Proof A sub-7i-model R C M x M which is an equivalence relation
in Set is obviously an equivalence relation in Modri (see 2.5.2). And
all equivalence relations in Modxi are of this type. Indeed from 2.5.2,
given an equivalence relation R >F in Modr, Modr{T{Ti^—)^R) is
an equivalence relation on Modr G'(ri, -), F), but by 3.3.3 this means
exactly that the 7i-model U{R) is an equivalence relation on the 71-
model U{F). D
Proposition 3.5.2 Let T be an algebraic theory. In Modr, equivalence
relations are effective and the functor U: Mod^? >Set preserves and
reflects the coequalizers of equivalence relations.
Proof We work with the corresponding presentation 71 of the algebraic
theory (see 3.3.4). Given an equivalence relation RC M x M in Modri
we consider the corresponding quotient q: M >M/R in the category
of sets. We shall prove that q underlies a 71-homomorphism.
Given an operation a € T{T'^^ T^) and elements Xi,..., Xn € M, we
realize a in M/R by
a([xi],..., [xn]) = [a(xi,..., Xn)].
This definition makes sense because given 2/1,..., 2/n ^ M such that
[xi] = [Vi]', i = 1,..., n, one has (xi^yi) G R for each index i and thus
(a(xi,..., Xn), aB/i,..., y^)) € R]
in other words,
[a(xi,...,Xn)] = [aB/i,...,2/n)].
It is obvious that M/R provided with these operations is a 71-model,
since M is, and q becomes in this way a 71-homomorphism.

140 Algebraic theories
Let us prove that the 7i-homomorphism q is the coequalizer of the
two projections pi,P2- ^ > ^ in Modri- If ^: M >L is a 7i-hom-
omorphism coequalizing pi,P25 we get in the category of sets a unique
mapping h: M/R >L satisfying hoq = g. We must prove that /i is a
7i-homomorphism. Given a € T{T'^^ T^) and xi,..., Xn € M,
/i^a([xi],..., [xn\)j = h[a{xi,..., x^)]
= /io^(a(xi,...,Xn))
= ^(a(xi,...,Xn))
= a(^(xi),...,^(xn))
= a{{h o q){xi), ...,{ho q){xn))
= a{h[xi\,.,,,h[xn\),
since ^ is a 7i-homomorphism.
We have already proved that the coequalizer of pi,P2- R ^M is
computed in the same way in Modri ^^^ i^ Set. Since kernel pairs are
also computed in the same way in Modri ^^d in Set (see 3.4.1) and
equivalence relations are effective in Set (see 2.5.5.a), equivalence relations are
effective in Modri •
We have already observed that the functor U: Modr > Set preserves
coequalizers of equivalence relations. Since moreover U reflects
isomorphisms (see 3.3.3), it reflects coequalizers of equivalence relations (see
2.9.7, volume 1). D
Corollary 3.5.3 Let T be an algebraic theory The forgetful functor
U: Mod^2 >Set preserves and reflects regular epimorphisms.
Proof Since Modr has kernel pairs (see 3.4.1), every regular epimor-
phism is the coequalizer of its kernel pair (see 2.5.7, volume 1), i.e. the
coequalizer of an equivalence relation (see 2.5.5.e). One derives the
conclusion by 3.5.2. D
Theorem 3.5.4 Let T be an algebraic theory The category Modr is
regular and exact in the sense of 2,1,1 and 2,6,1,
Proof Modr is complete and cocomplete (see 3.4.1 and 3.4.5) and
U: Modr >Set preserves and reflects pullbacks and regular
epimorphisms (see 3.4.1 and 3.5.3). Therefore regular epimorphisms are stable
m Modr under pullbacks, since they are in Set (see 2.4.2). Thus Modr
is regular and, by 3.5.2, exact. D

3.6 The algebraic lattices of subobjects 141
Corollary 3.5.5 Let T be an algebraic theory. The forgetful functor
U: Mod^2 >Set is exact in the sense of 2.3.5. D
3.6 The algebraic lattices of subobjects
We recall a classical definition
Definition 3.6.1 Let L be a complete lattice. An element k E L is
compact when k < \/i^jXi implies the existence of a finite subset J C I
such that k < \/^^jXi. An algebraic lattice is a complete lattice in which
every element is a join of compact elements.
Proposition 3.6.2 In an algebraic lattice, Rnite meets distribute over
filtered joins.
Proof Let us consider a filtered family {xi)i^i^ i.e. a non-empty family
such that
Vz, j El 3k E I Xi <Xk, Xj < Xfc.
Given an arbitrary element x of the lattice, we must prove the equality
xA{\fxi\= \/{x Axi).
\iei J iei
Prom the relations x A (Vie/ ^0 ^ ^ ^ ^i we deduce immediately that
X A l\J Xi\ > \J{x AXi).
Kiel I iei
It remains to prove the converse inequality. We can write
^A l\/xi \ = \/ kj
\iei J jeJ
for a family of compact elements {kj)j^j. Prom kj < Vie/^* ^^ deduce
that kj < \l^^i.Xi for some finite subset Ij C /. By filteredness of the
family {xi)i^i we choose ij E I such that Xi < xi. for every i E Ij. This
implies kj < Xi. and since kj < x, we deduce kj < x A Xi.. D
Proposition 3.6.3 Let T be an algebraic theory. The category Modr
is well-powered.
Proof ModT- is complete (see 3.4.1) and possesses a strong generator
(see 3.3.3); one derives the conclusion by 4.5.15, volume 1. D

142 Algebraic theories
Lemma 3.6.4 Let T be an algebraic theory and 71 its corresponding
presentation. Given a family {Mi C M)i^i of submodels of a Ti-model
M, the union [Ji^jMi in Modxi exists and is given by the set of all
elements a(xi,..., Xn) of M, with a an n-ary operation and xi,..., x^
elements of the set theoretical union of the subsets Mi C M.
Proof Every submodel M' C M which contains the M^'s must contain
the elements described. On the other hand, those elements obviously
constitute a submodel of M.
Proposition 3.6.5 Let T be an algebraic theory. The submodels of
every T-model constitute an algebraic lattice.
Proof We work with the corresponding presentation 71 of the algebraic
theory. It is obvious that a set theoretical intersection of submodels
is again a submodel; on the other hand lemma 3.6.4 gives an explicit
description of unions.
Given a subset X C M of a 71-model M, all the elements of the
form a(xi,...,Xn) for n € N, x^ € X and a an n-ary operation
obviously constitute a submodel X of M, called the submodel generated
by X. When X has finitely many elements Xi,...,Xn, this submodel
X is a compact element in the lattice of submodels. Indeed if the M^'s
are submodels and X C [Ji^jMi, each element Xk £ X has the form
CK(yi5 • • • 5 2/0 for some elements 2/1,..., y^ in the set theoretical union of
the Mi^s (see 3.6.4). Considering the finitely many elements Xk € X,
we get altogether finitely many elements yj in the set theoretical union
of the Mi^s and obviously those y^'s belong to the set theoretical union
of finitely many Mi's, say Mi,..., Mi. But the union Mi U ... U M; in
MocIt-j now contains the elements xi,..., Xn and therefore the submodel
X they generate.
Every submodel M' C M is obviously the union in Modr of all the
submodels of M' generated by a single element x € M', which concludes
the proof. D
Proposition 3.6.6 Let T be an algebraic theory. The forgetful
functor U: Modx >Set preserves and reHects arbitrary intersections and
Altered unions of subobjects. In particular, in Modr, finite intersections
distribute over Altered unions.
Proof Given a filtered family of subobjects M^ C M in Modr, we
consider the corresponding filtered colimit L = colimMi and the
induced factorization /: L >M. In the category of sets, U{L) is the

3,7 Algebraic functors 143
filtered colimit of the U{Miys (see 3.4.2), thus it is their union and
U{f): U{L) >U{M) is the canonical inclusion. Therefore / is a mono-
morphism (see 3.4.4) and L is the union of the M^'s. The rest follows
from 3.6.2, 3.4.1 and 3.4.2. D
3.7 Algebraic functors
Definition 3.7.1 Let K and T be algebraic theories, with objects
respectively written i2^, i2\ ..., i2^,... and T^, T\ ..., T"",.... A mor-
phism of algebraic theories is a functor F: TZ >T which preserves
finite products and maps R^ to T^ (n € N).
Observe that the requirement F{K^) = T^ does not follow from the
preservation of finite products. Indeed T^ is the terminal object of T so
that the constant functor on T^ preserves finite products.
Proposition 3.7.2 The theory of sets is an initial object in the category
of algebraic theories and their morphisms.
Proof We have seen in 3.3.5.b that the theory S of sets is (up to
an equivalence) the dual of the category of finite sets. In the category
of finite sets the only morphisms {*} >{1,... ,n} are the canonical
injections of the coproduct Ur=i{*}* Therefore in S the only n-ary
operations S'^ >S^ are the projections, from which follows the existence
of a unique morphism S >T for every algebraic theory T. D
Corollary 3.7.3 Let T be an algebraic theory and S the theory of sets.
The forgetful functor U: Mod^? >Set is the functor of composition
with the unique morphism of algebraic theories a: S >T.
Proof Given a T-model G: T >Set, the composite G o a preserves
finite products since G and a do. The set X corresponding to G o a is
G o (t{S^) = G{T^), i.e. the set UG] see 3.3.3. D
Definition 3.7.4 Let F: TZ >T be a morphism of algebraic theories.
The functor of composition with F
Modr >Modn, Gy-^GoF,
is called an algebraic functor.
Observe that this definition makes sense: indeed GoF preserves finite
products since F and G do.

144
Algebraic theories
Set ( ^^ MOVn
F*
Diagram 3.3
Modr
Proposition 3.7.5 An algebraic functor is faithful and rejects
isomorphisms; it preserves and reflects regular epimorphisms, coequalizers of
equivalence relations, small limits and filtered colimits.
Proof Consider a morphism of theories F: K >T. This yields the
commutative triangles of diagram 3.3 where S is the theory of sets,
Uui Ur are the forgetful functors and F* is the algebraic functor of
composition with F.
The functor F* is faithful since Ur is faithful; it reflects isomorphisms,
regular epimorphisms, coequalizers of equivalence relations, small limits
and filtered colimits since these are preserved by Un and reflected by
Ur- The functor F* preserves also regular epimorphisms, coequalizers
of equivalence relations, small limits and filtered colimits since these are
preserved by Ur and refiected by Un\ see 3.3, 3.4, 3.5. D
In order to prove that every algebraic functor has an adjoint, we need
to observe a property of the category of sets (or, more generally, of a
cartesian closed category; see chapter 6).
Lemma 3.7.6 Consider two functors F: A >Set and G: B >Set,
with A^ B small categories. The colimit of the functor
->Set
^ X B ^^^>Set X Set
is just (colimF) x (colimG).
Proof We have seen in 3.1.6. f, volume 1, that the functor Ax — admits
i—)^ as a right adjoint; therefore the functor Ax — preserves colimits
(see 3.2.2, volume 1). So
(colimF(A)) X {colim G{B)) ^colim{F{A) x colim G{B))
colimcolim(F(A) x G{B)) ^ colim{F{A) x G{B))
(A,B)
by associativity of colimits (see 2.12, volimie 1).
n

3.7 Algebraic functors 145
Theorem 3.7.7 Every algebraic functor has a left adjoint.
Proof Given a morphism F: K >T of algebraic theories and a model
G: K >Set of K, the reflection of G along the functor of composition
with F,
F*: Fun(T,Set) ^FunGe,Set),
is the left Kan extension K oi G along F (see 3.7.2, volume 1). To
conclude the proof, it suffices to verify that K preserves finite products.
We write respectively iZ^, i2\ ..., i2^,... and T^, T\ ..., T^,... for
the objects of Ti and T. Using the construction of 3.7.2, volume 1, we
know that K{T'^) is obtained as the colimit of the composite
Sn ^^^^-^n ^—^Set
where the objects of £n are the pairs {R^,t) with t: F{K^) >T''\
a morphism of £n r: {R^,t) >{R\s) is a morphism r: R^ >R^
in n such that 5 o F{r) = t. Since F{R^) =T'^, £n can equivalently
be described as the category having for objects the pairs (m, t) with
t: T^ >T^; a morphism r: (m, t) >(/,5) of £^n is a morphism in TZ
r: R^ >R^ such that 5 o F{r) = t. Essentially, we must prove that
KiT"") ^ colim(G o 0^) ^ (colimG o (/>i)^ ^ K{T^)''.
By using 3.7.6 and the preservation of finite products by G, we have
(colimG o (/>i)^ ^ colim(G o (/>i)^ ^ colimG o @1 )^
where ((/>i)^ stands for the composite
gix...xgi '^^^•••^'^iOlx...x7l ^ >Tl
mapping (mi,ti)i=i,...,n to R^^ x • • • x K^'^. On the other hand we can
consider the functor
Ipn- £i X ... X £i >£n
mapping (mi,ti)i=i^...,n to {Yl7=i'^i^lil7=i'^i)^ which is such that the
relation <t>n ^ '^n = @i)^ holds. To conclude the proof, it sufl^ices to
observe that the functor ipn is cofinal (see 2.11.1, volume 1).
Let us consider the following situation:
£ix...x £i —^i^^^-^£n S,—^x
where if is an arbitrary functor. Writing Ax for the constant functor
on the object X £ X, a, cocone a: H =^ Ax on H induces a cocone

146 Algebraic theories
a * ipn'- H o7pn=^ Ax on H o ip^- Conversely a cocone /?: H 01/;^=^ Ax
onHoipn has the form a^ipn for a unique cocone a: H => Ax- Indeed,
given (m, t) € £n the diagonal 8: K^ >{K^)'^ yields a morphism of
£n
6: {m,t) > (^ >< ^'11^=/^ ^ V '='^^((^'^^^^)^=iv..,n)
so that it is necessary to define
The naturaUty of C impUes immediately that of a. Thus the cocones on
H are in bijection with the cocones on H o Tp^, from which it follows
easily that ipn is a cofinal functor, as in 2.11.2, volume 1. D
Notice that the second condition for cofinality, as described in 2.11.2,
volume 1, has no reason to be satisfied in the previous situation. It was
used in 2.11.2, volume 1 to prove the equivalence between the various
possible ways of extending the original cone. In the situation of 3.7.7,
the diagonal morphisms offer a canonical way to realize the extension.
Corollary 3.7.8 Let T be an algebraic theory. The forgetful functor
U: Mod^2 >Set has a left adjoint.
Proof By 3.7.7 and 3.7.3. D
Observe that 3.7.8 could already have been obtained from 3.3.3 and
3.4.5: the singleton admits T{T^, —) as a reflection along U from which
lIx€X^(^^' —) is the reflection of the set X along J7; see 3.8.3, volume 1.
3.8 Freely generated models
Definition 3.8.1 Let T be an algebraic theory, U: Modr >Set the
corresponding forgetful functor and F: Set >Modr its left adjoint.
A) By a free T-model we mean, up to isomorphism, a model of the form
F{X) for some set X;
B) by a finzteiy generated T-model, we mean a model M which is a
quotient of a free model F{n) on a finite set n;
C) by a Gnitely presentable T-model, we mean a model M which can
be obtained via a coequalizer diagram
F{m) ^F{n) »M
where m, n are finite sets.

3.8 Freely generated models 147
Proposition 3.8.2 Let T bean algebraic theory. The free model functor
F: Set )>Modr preserves monomorphisms.
Proof Let i: X > >Y be an injection in Set. If X is not empty, i admits
a retraction r: Y >X and from roi = Ix, we get F{r)oF{i) = 1f(X)
proving that F{i) is a monomorphism.
If X = 0 is the empty set, i.e. the initial object of Set, F@) is
the initial object of Modr; see 3.2.2, volume 1. If t/F@) is empty,
t/F@) ^llf'iY) is injective and thus F@) >F{Y) is a
monomorphism; see 3.4.4. If UF{^) is not empty, every mapping Y >t/F@)
induces by adjunction a T-homomorphism t: F{Y) >-F@); see 3.1.1,
volume 1. The relation t o F{i) = 1/^@) holds since F@) is an initial
object, thus F{i) is a monomorphism. D
Lemma 3.8.3 Let T be an algebraic theory. With the previous no-
tationSy every free model F{X) can be written as a filtered colimit
F{X) = colimnCX-P(^) where n runs through the finite subsets of X.
Proof Obviously X = coUmncx^ holds in Set and F preserves coUmits
(see 3.2.2, volume 1). D
Lemma 3.8.4 Let T be an algebraic theory. With the previous
notations, the finitely generated free models are exactly, up to isomorphism,
the models F{n) for n a finite set.
Proof Suppose F{X) is a quotient of some F(n), for n a finite set, and
consider the corresponding regular epimorphism q: F{n) »F{X). We
can write F{X) = coliuijncxF{m) where m runs through the finite
subsets of X. By construction of filtered coUmits in Modr (see 3.4.2),
for every index i the element U{q){r]n{i)) arises from an element in some
UF{mi)^ for rrii C X, m^ finite. Since n is finite, all those elements are
already in some UF{m), ior fi: m ^^ X a. finite subset of X, namely
the union of the m^'s. This yields a mapping u: n >UF{m) and thus
a corresponding factorization v: F{n) >F(m), since (^F{n),rjn) is the
reflection of n along U. Diagram 3.4 indicates already that U{v)orjn = u.
Therefore
UF{ii) o U(y) or]n = UFifi) ou = U{q) orjn
and q = F{fi) o i; by the uniqueness condition in the definition of a
reflection; see 3.1.1, volume 1.
Since g is a regular epimorphism, F(/x) is a regular epimorphism as
well (see 3.5.4 and 2.1.6). But since /x is a monomorphism, so is F{/jl)
(see 3.8.2). Finally, F{fi) is an isomorphism (see 3.5.4). D

148 Algebraic theories
n —y^-^UF{m)
Vn
UF{i)
UF{n)-jj^>UF{X)
Diagram 3.4
Proposition 3.8.5 An algebraic theory T is equivalent to the dual of
the full subcategory of Modr whose objects are the finitely generated
free models.
Proof With the previous notation, we know that the free model F{n)
on n generators is just T{T'^, —); see 3.3.3. By the Yoneda lemma (see
1.3.3, volume 1)
ModT{F{n),F{m)) ^ Nat(T(T^,-),T(T^,~)) ^ T(r^,T^). D
Proposition 3.8.6 The free models of an algebraic theory T are
projective.
Proof Consider a free model F{X)^ a strong epimorphism p: M » AT
in Modr and a morphism /: F{X) >N. The set theoretical mapping
U{p): U{M) »U{N) is a surjection (see 3.5.3, 3.5.4 and 2.1.4); thus,
by the axiom of choice, it has a section s: U{N)> >U{M). By
adjunction, / corresponds to a morphism g: X >U{N) and the composite
sog: X >U{M) to a morphism h: F{X) >M. Let us prove that
p o /i = /. By adjimction, this is equivalent to U{p) o so g = g^ which is
the case since 5 is a section oiU{p). D
We keep the notation U: Modr >Set, F: Set > Modr for the
forgetful functor and the free model functor. We write rj: Ig^^ => UF and
e: FU => iModr ^^^ ^^^ natural transformations of the adjimction (see
3.1.5, volume 1). We recall they make commutative the triangles of
diagram 3.5.
Lemma 3.8.7 With the previous notation, £m isa regular epimorphism
for every T-model M.
Proof By the first triangle, U{eM) is surjective and thus sm is a regular
epimaphism (see 3.5.3). D

3.8 Freely generated models
149
r)*U
U —'■ >UFU
U *s
U
F *ri
F '-^FUF
e*F
Diagram 3.5
UFU{M)-—^UFU{N)
U{sm)
U{eN)
U{M)-
-^U{N)
f
Diagram 3.6
Proposition 3.8.8 Let T be an algebraic theory and M, N two T-
models. A mapping f: U{M) >U{N) has the form U{g) for a
(necessarily unique) T-homomorphism g: M >N iff the square in
diagram 3.6 conmiutes.
Proof If / = U{g)^ the commutativity holds by naturality of e.
Conversely, suppose the commutativity of the given diagram. In Modr
consider diagram 3.7, where the horizontal lines are kernel pairs. Applying
t/, we get
U{eN) o UF{f) o U{ui) = f o UisM) o U{u,)
= foU{eM)oU{u2)
= U{eN)oUF{f)oU{u2).
This yields eNoF{f)oui = eN^F{f)ou2 by faithfulness of U (see 3.3.3)
and therefore a unique factorization h such that F{f) o m = Vi o h, i =
1,2.
Now £n, £m are regular epimorphisms (see 3.8.7), so are the coequaUz-
ers of their kernel pairs; see 2.5.7, volume 1. The relation eNoF{f)oui =
En o F{f) o U2 thus implies also the existence of a unique g: M >N
such that go€M = sn ^^if)- Applying U we get
U(g) o U(eM) = U(eN) o UF(f) = / o U{eM)

150
Algebraic theories
P ]FU(M) ^ » M
U2
Q
Vi
F{f)
V2
%FU{N)-
sn
Diagram 3.7
-» N
from which U{g) = /, since U^sm) is surjective.
D
Proposition 3.8.9 Let T he an algebraic theory. Every T-model M
is a quotient of a free model. More precisely the following diagram is a
coequalizer:
^FU{M)
FUFU(M) I FUiM) —^^^^—» M.
FUisM)
Proof We know aheady that sm is a regular epimorphism (see 3.8.7),
thus M is a quotient of the free model FU{M) and sm is the coequalizer
of its kernel pair (see 2.5.7, volume 1).
To get the stated canonical coequalizer let us observe the equality
^M o^FU{M) = ^M o FU{sm)^ which follows just by naturality of e. Now
given h: FU{M) >N in Modr such that h o SpuiM) = ho FU{eM)^
we must construct g: M >N such that g o sm = h\ since sm is an
epimorphism, g will necessarily be unique. By adjunction, h corresponds
to a mapping /: U{M) >U{N) such that the relations eN^F{f) = h
and / = U{h) o r]u(^M) hold. By naturality of ry, the triangular identities
of the adjunction and the assumption on /i,
/ o U{£m) = U{h) o r]u(M) o U{eM) = U{h) o UFU{eM) o r]uFU{M)
= U{h) o U{eFU{M)) o VuFUiM) = U{h) = U{eN) o UF{f).
Therefore the mapping / underlies a T-homomorphism g: M > AT; see
3.8.8. FVom U{g) = / we get U{g) o U{eM) = U{h), thus ^ o ^m = ^ by
faithfuhiess of U\ see 3.3.3. D
Proposition 3.8.10 Let T he an algebraic theory The finitely
generated free models constitute a family of dense generators of Modr (see
3.5.4, voJizme 1).
Proof Modr is reflective in Fun(T,Set); see 3.4.5. In Fun(T,Set),
every T-model F € Modr is the colimit of the canonical diagram of all

3.8 Freely generated models
»M
151
1 _!Z3—^C/FA)-^^G(M)
U{h)
Diagram 3.9
representable functors T{T'^^ —) over F (see 2.15.6, volume 1). Since the
representable functors are already in Modx, this is a colimit in Modx
and one gets the conclusion by 3.3.3D). D
Proposition 3.8.11 Let T be an algebraic theory Every T-model is
the filtered union of its finitely generated T-submodels.
Proof We use the notation of 3.8.10. For each object {F{n), u) of #'/M,
we consider the image of u (see 3.5.4 and 2.1.4) as in diagram 3.8. Each
lu is, by construction, a finitely generated submodel of M. The smallest
submodel of M containing all the luS is M itself since, given an element
X € U{M)^ we can consider diagram 3.9. In fact x = U{x) o ?7i(*), thus
X € U{I-x) and therefore the set theoretical union of the U{Iu) contains
already all the elements of U{M).
It remains to prove that the union is filtered. Given finite sets n,m
and morphisms u: F{n) >M v: F(m) >M, let us consider the
corresponding factorization
(w, v): F{n -h m) = F{n) II F{m)-
->M.
Observing diagram 3.10, we deduce that lu Q I{u,vy'> see 4.4.5, volume 1.
In the same way ly C I(^u,v)^ which proves the filteredness. D
It should be observed that even if the construction of the previous
proof is based on that of 3.8.10, the category ^/M in 3.8.10 is not
filtered. Indeed, two parallel morphisms in #"/M have no reason to be

152
Algebraic theories
F(n-f m) »I{u,v)> > M
F{n)-
^> lu >-
Diagram 3.10
-> M
coequalized in ^jM. In 3.8.11 this difficulty vanishes since between
subobjects, we just take as morphisms the canonical inclusions.
Now let us turn our attention to the case of finitely presentable models.
If we have such a model M presented as a coequalizer
F(m)\
u
-%F(n)-
^>M
with m, n finite, u and v correspond by adjunction to mappings in Set
%v: m \TTF{n)^ thus to m-tuples (wi,..., Wm), (i^i,...,Vm) of
elements of UF(n), It is immediate that M is the quotient of F{n) by the
smallest congruence containing the pairs (ui^Vi), for i = 1,... ,n. Thus
M is the model obtained from:
• finitely many generators: the elements of n;
• finitely many relations Ui = vi between terms constructed from those
generators (see 3.2.6).
This justifies the terminology "finitely presentable".
Proposition 3.8.12 Let T he an algebraic theory. The finitely
presentable models constitute a dense family of generators of Modx, stable
under finite colimits. Moreover every T-model is a filtered colimit of
finitely presentable ones.
Proof We write ^ for the full subcategory of finitely presentable T-
models. Given a model M, we consider as in 4.5.4, volume 1, the category
^/M and the forgetful functor (j): ^/M >Modr- We consider the
canonical cocone
S{p,u)' (t>{P^u)-
->M, s^p^u) = u,
where P € ^ and u: P-
this cocone.
►M. We must first prove the universality of

3.8 Freely generated models 153
Given another cocone on 0,
hp,u)' (t>{P.u) >N
consideration of the morphisms t(^F{n),u) yields, by 3.8.10, a unique
factorization g: M >N such that g o S(ir(n),w) = i{F{n),u) for every finite
set n and morphism w. F{n) >M. Now given P £ ^ and u: P >M,
we consider a regular epimorphism p: F{m) ^>P with m finite. This
yields a morphism p: (F(m), wop) >(P, u) in ^/M and the relations
^(P,w) op = t(F(m),uop) =9^ «(F(m),wop) =9^ ^(P,w) ^ P
imply t(^p^u) = 9 ^ ^{P,u) since p is an epimorphism.
We have already proved that the finitely presentable objects are a
dense family of generators. It remains to prove that ^/M is filtered.
This will certainly be achieved if we prove that ^ is stable in Modr for
finite colimits.
Now ^ contains the initial object F@) and, if P, Q are finitely
presentable via
F(m) " ]F{n) ^-^>P, F(fc) [ If{1) i-^>Q,
it follows from 2.12.1, volume 1, that
F{m)UF{k)ZZ ^P(n)UP(/) ^"^ ^PUQ
vUs
is still a coequalizer. This proves that PIIQ is finitely presentable, since
F{m) n F{k) = F{m + k) and F{n) U F{1) = F{n + /).
Finally ^ is stable under coequaUzers. Consider diagram 3.11, where
the columns are coequalizers in Modr as well as the bottom row. The
sets kj^m^n are finite, so that P,Q are finitely presentable. We must
prove that R is finitely presentable as well. By projectivity of F{n) with
respect to regular epimorphisms (see 3.8.6), we choose morphisms x^y
such that q o X = wop, q o y = i;op. We consider then the object
F{n) II F{1) = F{n -f /) and the corresponding factorizations x II c,
yUd. We shall prove that roq = Coker(xIIc,yIId), showing that R is
finitely presentable. First r o q coequalizes x, y, since r coequaUzes w, v
and r o q coequalizes c, d since q coequalizes c, d. Next, if a morphism
z: F{k) >Z coequalizes xllc, ylld, z factors as moq since zoc = zod.
Now
'mouop=zrnoqox = zox=^zoy = moqoy=:Tnovop

154
F{m)
Algebraic theories
F{1) Fin + l)
Diagram 3.11
from which mou = mov^ since p is an epimorphism. Therefore m factors
as n o r. D
We pursue this section with charaterizations of the finitely generated
and finitely presentable models.
Proposition 3.8.13 Let T be an algebraic theory. A T-model M is
Gnitely generated precisely when the representable functor
Modr(M,-): Modr >Set
preserves Altered unions.
Proof Suppose M is finitely generated and choose a regular
epimorphism q: F{n) ^M, with n finite. Given a filtered family of submodels
Ni Q N^ their filtered union is just the corresponding filtered colimit and
it is computed as in the category of sets (see 3.6.6). We must prove the
isomorphism
U.^^Modr(M, AT,) ^ Modr (^M,\J^^^Ni) .
Clearly the left-hand side is contained in the right. Conversely given
/: M ^Ui€/^*' ^^^ composite foq: F{n) ^Uie/^* corresponds by
adjunction to a mapping n >U (Ui€/^0' ^^^^ ^^ ^ family xi,..., Xn
in U {[Ji^jNi) of elements. Since U preserves filtered unions, each Xk is in
some U(Nif^) and by filteredness, there exists an zq such that xi,..., x^
are in U^Niq). This means that foq factors through Ni^. But since U{q)
is surjective, this implies that U{f) takes all its values in [/(AT^q), thus /
factors through Ni^. Therefore / is in the left-hand side of the required
isomorphism.

3.8 Freely generated models 155
Now suppose the previous isomorphism holds for every filtered union.
We apply it to the filtered union of the finitely generated submodels Mi
of M, which is M itself (see 3.8.11):
u,
Modr(M, Mi) ^ Modr(M, M).
The identity on M corresponds, in the left-hand side, to some morphism
/: M >Mi for some finitely presentable submodel Mi. In other words,
the composite
M i- >Mi CM
is the identity on M. In particular the inclusion M^ C M is a strong
epimorphism, thus an isomorphism (see 4.3.6, volume 1). Thus M is
isomorphic to the finitely generated model Mi. D
Proposition 3.8.14 Let T be an algebraic theory. A T-model M is
finitely presentable precisely when the representable functor
Modr(M,-): Modr >Set
preserves filtered colimits.
Proof Let us consider a filtered colimit N = colimAT^ in Modr- The
canonical morphisms sf. Ni >N induce a cocone Modr(M, Si) thus a
canonical factorization
(p: colim Modr(M, Ni) >Modr(M, colimNi).
Let us assume that M is finitely presented by the coequalizer
u
F{m) > F{n) 2—^> m
and let us prove that tp is a. bijection.
Given a morphism /: M > coUm AT^, we get a composite morphism
/ o q: F{n) > colim AT^, thus by adjunction a corresponding mapping
?i >{/(colim A/^), thus finally ?2-elements Xi,... ^Xn £ U{colimNi).
Since U preserves filtered colimits (see 3.5.2), those elements Xj can be
written as the equivalence classes of elements xJ € U^Ni^) (see 2.13,
volume 1; by filteredness, there is no restriction in supposing all the xJ
in the same U^Ni^). This yields a new mapping n >U{NiQ) and thus
a corresponding morphism g: F{n) ^Ni^ such that Si^o g = f oq.
The morphisms gou\ F{m) ^Ni^^ gov: F{m) >NiQ correspond
in the same way to families 2/i,..., 2/m and 2:1,..., z^n of elements of
U{NiQ). The equalities
SiQogou = foqou = foqov = SiQogov

156 Algebraic theories
imply the corresponding equalities [yk] = [zk] betwen equivalence classes
in the coUmit. Since there are just finitely many identities of this kind,
by filteredness, they are already realized at some further level Ni^ of the
colimit (see 2.13.4, volume 1). Thus there is a morphism s: Ni^ >Ni^
in the diagram defining the colimit such that s{yk) = s{zk) for every k.
By adjunction, this implies now so gou = so gov.
Considering the relation sogou = sogov^we now get a unique
factorization h: M >Ni^ through the coequalizer q of w, v; thus hoq =
sog. This immediately implies
Si^ohoq = Si^osog = SiQog = foq^
from which Si^ o h = f since q is an epimorphism. This means precisely
that h € Modr(M, Ni^) and f = ^{[h])^ proving that (p is surjective.
The injectivity of cp is analogous and even easier. Given two morphisms
h € Modr{M,NiJ and k £ Modr{M,Ni^) such that ip{[h]) = (/?([fc]),
we can choose a further level Ni^ and assume h and k are defined at this
level Ni^. The two morphisms ho q^k o q: F(n) I N^^ correspond to
families of elements ai,..., a^ and 6i,..., 6^ in U[Ni^). The relations
[oLj] = Xj = [bj] hold in the colimit since (/?([/i]) = </?([fc])- Therefore,
by filteredness, all the relations [aj] = [bj] are already realized at some
further level AT^^, for some morphism s': Ni^ >Ni^ in the diagram.
These relations, by adjunction, mean precisely s'ohoq = s'okoq^ thus
s'oh = s'ok since q is an epimorphism. But this means exactly [h\ = [fc],
proving the injectivity of (/?.
We must prove now the converse implication. We suppose that ip is
an isomorphism for every filtered colimit. We apply this assumption to
the case of the filtered colimit of 3.8.12 defining M itself:
if: colimModT(M,(/>(P,w)) ^ UoAt{M,M).
The identity on M corresponds to the equivalence class of some
morphism /: M >P, on the left-hand side; in other words, the composite
M ^-—>P = (t>{P,u) ^^^-^^ >M
is the identity on M. Thus, M is a retract of P and therefore the following
diagram is a coequalizer:
(see 6.5.4). Since finitely presentable objects are stable under finite co-
limits (see 3.8.12), M is finitely presentable. D

3.8 Freely generated models 157
Let us conclude this section with some useful observations related with
the consideration of finitely generated free modules on a ring.
Proposition 3.8.15 Let R he a ring with unit and TZ the theory of
right R-modules. The free R-module on n generators (n € N) is just the
power R^ with the pointwise operations.
Proof If M is a right i?-module, one immediately gets bijections
Modi2(i2, M)^M^ Set({*}, M)
by mapping a linear mapping /: R >M to /(I) E M and an element
m € M to the linear mapping
g-Jl >7Vf, ry-^mr.
Thus R is the free right i?-module on a single generator. Since n =
1II... II1 in Set and the free module functor preserves coproducts (see
3.2.2, volume 1), the free i?-module on n generators is the n-th copower
of i?, which is R^ since the category of modules is abelian (see 1.4.6.a
and 1.6.5). D
Proposition 3.8.16 Let Rhea ring with unit and TZ the theory of right
R-modules. Then the dual category TZ* is the theory of left R-modules.
Proof Clearly TZ is the dual of the full subcategory of right i?-modules
generated by the objects R^] see 3.8.5 and 3.8.15. Since R^ is both an
n-th power and an n-th copower, both TZ and its dual TZ* are algebraic
theories. Since by 3.8.15 R^^ provided with the pointwise operations (on
the left), is the free left i?-module on n-generators, we must prove the
existence of an isomorphism
i2-Mod(i2^,i2^) ^ Modi2(i2^,i2^),
where i?-Mod indicates the category of left modules, and Mod/^ the
category of right modules. But since the modules involved are free, we can
write a left i?-linear mapping R^ >R^ as
where {aij)ij is an n x m matrix. In the same way a right i?-linear
mapping R^ >R^ can be written as
xi \ / xi

158 Algebraic theories
where {aij)ij is again a n x m matrix. This obviously yields the required
isomorphism and clearly it acts contravariantly with respect to the
composition, n
3.9 Characterization of algebraic categories
We want now to characterize those categories which can be presented as
categories of models for some algebraic theory.
Theorem 3.9.1 Let ^ be a category and U: ^ ^'Set a functor. The
following conditions are equivalent:
A) ^ is equivalent to the category of models of some algebraic theory
T, with U the corresponding forgetful functor;
B) the following conditions are satisfied:
(a) ^ has coequalizers and kernel pairs;
(b) U has a left adjoint F;
(c) U reflects isomorphisms;
(d) U preserves regular epimorphisms;
(e) UF preserves filtered colimits.
Under these conditions, T* is equivalent to the full subcategory of ^
generated by the objects F(n), for n running through the finite sets. A
category ^ as in the statement is called an ^^algebraic category".
Proof By 3.4.5, 3.4.1, 3.7.8, 3.3.3 and 3.5.3 conditions B)(a), (b), (c),
(d) are necessary. By 3.2.2, volume 1, and 3.4.2, this volume, condition
B)(e) is necessary as well. The very last assertion follows from 3.8.5.
Conversely, if conditions B) (a) to B)(e) are satisfied, U preserves
kernel pairs (see 3.2.8, volume 1). Since U reflects isomorphisms, it also
reflects kernel pairs (see 2.9.7, volume 1). This implies that U reflects
regular epimorphism. Indeed consider /: A >B in ^ such that U{f)
is a regular epimorphism in Set. Write (w, v) for the kernel pair of /.
Then {U{u), U{v)) is the kernel pair of U{f) and since U{f) is a regular
epimorphism, it is the coequalizer of (C/(w), t/(i;)); see 2.5.7, volume 1.
If q: B >Q is the coequalizer of (w, i;) and w: Q ^B the unique
morphism such that w o q = f^ then (w, v) is the kernel pair of q; see
2.5.8, volume 1. Therefore U{q) is another regular epimorphism with
kernel pair {U{u), U{v)), thus U{q) is another coequalizer of {U{u), U{v));
see 2.5.7, volume 1. Finally U{w)^ and therefore w, is an isomorphism,
proving that /, isomorphic to ^, is a regular epimorphism.
Writing 0,1,2,..., n,... for a specific choice of finite sets, we define T
to be the category with formally distinct objects T^, r\ T^,..., T^,...

3.9 Characterization of algebraic categories 159
MoAr
and T(T'^,T'^) ^ ^(F(m),F(n)). The composition is determined by
the fact that T is equivalent to the dual of the full subcategory of ^
generated by the objects F@), F(l), FB),..., F(n),.... By 3.2.2,
volume 1, F{n) is the n-th copower of F(l), so that T is indeed an
algebraic theory. For every object C E ^^ the contravariant representable
functor ^(—,C) transforms finite coproducts into finite products (see
2.9.5, volume 1), thus induces a T-model
<^(-,C): T >Set, T"" ^ ^{F{n),C),
Every arrow / € T{T'^,T'^), i.e. every arrow /: F{m) >F(n) in ^,
induces a corresponding mapping
<^(F(n),C) ><^(F(m),C), g^gof,
so that finally we have constructed a functor
(/?:^ >Modr, Ch^^(-,C), /^-^^(-,/),
and it remains to prove it is an equivalence of categories. To avoid
confusion, we write V: Mod^? >Set for the canonical forgetful functor and
G: Set >Modr for its left adjoint, as in diagram 3.12.
First of all, let us observe that (see 3.3.3)
{V o ^)(C) ^ ^(F(l), C) ^ Set(l, U{C)) ^ t/(C),
from which Vo(p^U. Observe also that for every finite set n (see 3.3.3)
(p{F{n)) ^ ^(-, F{n)) ^ T{T'', -) ^ G{n).
More generally, every set X can be written as a filtered colimit X =
colimXi where the Xi's run through the finite subsets of X, Since F
preserves colimits (see 3.2.2, volume 1), F{X) = colimF(Xi). We
consider the canonical factorization (see 2.12, volume 1)
ax: cohm(p{F{Xi)) >(p{co]imF{Xi)).

160 Algebraic theories
To prove that ax is an isomorphism, it suffices to prove that V{ax) is
an isomorphism (see 3.3.3). But since V preserves filtered colimits (see
3.4.2)
V(colim(p{F{Xi))^ ^ colim{Vocp){F{Xi)) ^ colim{U o F){Xi)
^{Uo F)(colimX^) ^ U{colimF{Xi))
^F(^(colimF(X,)))
from which the fact that V{ax) and thus ax is an isomorphism. In
other words, ipo F = G.
Now let us write rj: Ig^^ => UF^ e: FU => 1<^ for the natural
transformations associated with the adjunction F -\ U; see 3.1.3, volume 1.
Prom the relation {U*s)o{r]*U) = lu we deduce that U{ec) is a surjec-
tion for every C € ^, thus Sc- FU{C) ^C is a regular epimorphism;
let us write Sc = Coker(wc,i^c)- In particular considering the natural
bijections (see 3.1.3, volume 1)
^{FU{C),D) ^ Set{U{C),U{D))
and morphisms f^g: C \D^ the two composites / o sctQ o sc ^
'€{FU{C),D) correspond to U{f),U{g) € Set{U{C),U{D)) so that
U{f) = U{g) => foec = goec ^ f = g
since ec is an epimorphism. In other words, U is faithful and, since
U = Vo(p^ (f is faithful as well.
Let us also write a: Ig^^ => VG and r: GV => ^Modr ^^^ ^^^ ^^^"
ural trasformations associated with the adjunction G -\ V (see 3.1.3,
volume 1). The same argument as above (or as in 3.8.7) indicates that
Tm- GV{M) ^M is a regular epimorphism for each T-model M.
In order to prove that cp is full, consider first a set X and an object
-D G ^. By the various adjunctions and what is already proved we have
«'(F(X), D) ^ Set(X, U{D)) ^ Set(X, {V o cp){D))
^Modr{G{X),(p{D))
^Modr((^oF)(X),^(D)).
The definitions of the individual isomorphisms indicate that the
composite bijection is indeed that mapping a morphism /: F{X) >D to
(fiif). It remains now to replace F{X) by an arbitrary object C e^.

3.9 Characterization of algebraic categories 161
So let us choose C^D £^ and a morphism a: (p{C) => (p{D) in Modr-
We know already that the composite
^{FU{C)) ^^^^^ >cp{C) ^—>^(D)
has the form (p{f) for a unique /: FU{C) >D, We observe that, still
writing Sc = Coker (wc^c),
(p{f) o(p{uc) = ao(p{sc) o(p{uc) = cxo(p{ec) o(p{vc)
from which fouc = fovc, since cp is faithful. This implies the existence
of a unique factorization g: C ^D through Sc — Coker (wc^c)? thus
yielding g o sc = f-l^ particular
{V o cp)ig) o U{ec) = U{g) o U{ec) = U{f)
= {V o cp){f) = V{a) o {V o ^){ec) = V{a) o U{ec).
from which {V o (p){g) = V{a) since U{ec) is surjective and (p{g) = a
since V is faithful. This concludes the proof that cp is full and faithful.
Next we shall prove that (p has a left adjoint ip. For every T-model
M, we consider the kernel pair of tm^
R(M) I GV{M) —^^^^—» M.
sm
Since tm is a regular epimorphism, tm = Coker (rM^^M) by 2.5.7,
volume 1.
Let us consider the following composites:
cpFVR{M) ^ GVR{M) ^^^^^ >R{M)—^^^-^GF(M) ^ (pFV{M)
cpFVR{M) ^ GVR{M) ^^^^^ ^ R{M) '-^^—^GV{M) ^ ipFV{M).
Since cp is full and faithful, we get uniquely determined morphisms
ci'MibM such that
^{(im) = rM ^ trm, (p{bM) = Sm ^ trm-
We define qm = Coker (aM^^M),
CiM
FVR{M) I FV{M) —2M_^> ^(M).
bM
We consider now the kernel pair of qmi
rriM
S{M) :XFViM)—2M^>^(M).
TlM

162 Algebraic theories
GVR{M)^^^^R{M)ZI^ZXGV{M) ^^ » M
sm
7m:
(firriM)
(p{nM)
Diagram 3.13
^M
c^5(M)ZZ:tc^FF(M)——^>(^V^(M)
Since qm is a coequahzer, qm = Coker(mM5^M); see 2.5.7, volume 1.
Since U preserves kernel pairs and regular epimorphisms while V
reflects them, the image of this diagram under (p produces the kernel pair
{(p{mM)^^{'f^M)) of the regular epimorphism (p{qM)] as a consequence,
^{qm) is still the coequahzer of {^{mM)^^{'nM))'
In Modr we can now consider diagram 3.13, where the horizontal lines
are both kernel pairs and coequalizers. The relations
(p{qM) orMO Tr(m) = ^{Qm) o (p{aM) = ^{qm) o ^(hu)
= ^{qm) osm^ t'r(M)
imply (p{qM) o ^M = ^{qm) o sm^ since trm is an epimorphism. This
yields a unique factorization 7m- RM xpSM such that
(p{mM) o 7m = ^M, ^{nu) o 7m = sm-
Since the upper line is a coequahzer, we get a unique factorization mor-
phism 6m'' M >{(p o ip){M) such that 6m ^tm = ^{qm)-
Let us prove that {^{M)^6m) is the reflection of M along (p. Given
C e^ and a: M => (/?(C), the composite
cpFV{M) ^ GV{M)—2vi_^M ^—>(p{C)
has the form cp{x) for a unique x: FV{M) >C, since (p is fuU and
faithful. One has immediately
(p{x) o cp{aM) = a o tm o tm o tr^m) = olotmosm^ tr(^m)
= (p{x)o(p{bM),
from which xoaM = xoBm^ since the functor (p is faithful. And as ^m =
Coker (aMj^M), we get a unique y: ip{M) >C such that y oqm = x.
One has (p{y) o 6m = ot since
^{y) o6motm =" <p{y) o (p{qM) = ^{x) = a o TM

3.9 Characterization of algebraic categories 163
and tm is an epimorphism. We know y must be unique since 6m is an
epimorphism and cp is faithful.
Thus ip extends in a functor ip: Modq >^ which is left adjoint to
the full and faithful functor (/?; see 3.1.3, volume 1. In particular ip o cp
is isomorphic to the identity on ^ (see 3.4.1, volume 1) and it remains
to prove that the natural transformation 6: l^odr =^ ^ ^ ^ is itself an
isomorphism.
Since the pair (r^, sm) is a kernel pair, it is jointly monomorphic and
therefore jm is a monomorphism. More explicitly, if 7m o h = jm o fc,
then
rMoh = (p(rnM) o^M^h = (p{mM) o7m o ^ = ^m o fc,
SMoh = (p{nM) ojMoh = (^(um) ojMok = SMok,
from which h = k, since (rMiSM) is a pullback. Thus 7m is indeed a
monomorphism.
Since ^|; is left adjoint to (/?, consideration of 7m*- R{M) >(pS{M)
yields the existence of a unique morphism /im* ipR{M) >S{M) such
that (p{hM)^6ii(^M) — 7m- As 7m is a monomorphism, bjn^M) is a
monomorphism as well. But since (p{qR(M)) is a regular, thus strong
epimorphism, and (p{qR(M)) = ^r{M) o '7"ii(M), ^r{M) is also a strong
epimorphism and therefore an isomorphism (see 4.3.6, volume 1).
Consider now the diagram
(^{rriM o hu)
ip^ljR(M) ]GV(M) ^ »M.
cp{nM o Km)
Composing with the isomorphism 6rm we get
(p{mM o hu) o 6rm = ^{rriM) o 7m = ^m,
(firiM o ^m) o Srm = (fi^M) o 7m = Sm,
from which the above diagram is both a kernel pair and a coequalizer.
Next let us observe that
ttM = rriM ohM o qrm, bM = tim oKm ^ qRM-
We prove the first equality
(p{mM ohM o qrm) = (firriM) o cp{hM) o 6rm o trm
= ^{rriM) o 7m o trm = rMO trm

164 Algebraic theories
Modr — >Mod7^
from which we get the required result, since (p is faithful. Therefore
qM = Coker (ttim oHmo qrm, ^m o ^m o Qrm)
and since qrm is an epimorphism
Qm = Coker {ttim o ^m, ^m o ^m)-
But applying (p to the diagram
yields on the left-hand side the kernel pair of tm, as already observed.
Since kernel pairs are preserved by V and reflected by J7, the composite
pair {niM o ^m? ^m o ^m) is a kernel pair in ^, thus it is the kernel pair
of its coequalizer qm (see 2.5.8, volume 1).
Now (p preserves kernel pairs, since it has a left adjoint (see 3.2.2,
volume 1). Thus the previous diagram has for image the kernel pair
of ^{qm)i with ^{qm) a regular epimorphism since U preserves regular
epimorphisms and V reflects them. Thus (pqM is also the coequalizer of
the pair {<p{mM^hM)^ </^(^mo/im)), i-e. it is isomorphic to tm- In other
words, 6m is an isomorphism. D
Proposition 3.9.2 Let T, 72, be algebraic theories, with corresponding
forgetful functors f/, V and free algebra functors F, G, as in diagram 3.14,
A functor W: Mod^^ >Mod7^ is the algebraic functor induced by a
morphism of theories H: Tl >T if and only ifV oW = U,
Proof Let us use the notation
jy:
a:
lSet=^
lSet=^
UoF,
F<
>G,
e:
r:
Fo
:G<
'^=^lModr'
= V =» iModTe
to denote the natural transformations associated with the two
adjunctions F H f/, G H F; see 3.1.3, volume 1.

3.9 Characterization of algebraic categories 165
For every finite set n, the mapping ry^- n >UF{n) = VWF{n)
corresponds by adjunction with a morphism p^: G{n) >WF{n). We
shall prove that (F(n),pn) is the reflection of G{n) along W. Indeed,
given M € Modr and /: G{n) >W{M), we get by adjunction a
mapping g: n >FW(M) = U{M) and thus a morphism h: F{n) >M.
The relation W{h) o p^ = / is equivalent to VW{h) orjn = g^ thus to
U{h) orjn = g which holds by definition of h. Those relations prove at
the same time the uniqueness of h.
So each G{n) admits (F(n),pn) as reflection along W. As in 3.1.3,
volume 1, this extends to a functor between the full subcategories
generated by those objects: given an arrow /: G{n) >G{m)^ it is mapped
to the unique arrow h: F{n) >F{m) such that p^ o f = W{h) o p^.
Considering 3.8.5, T is the dual of the full subcategory of Modr
generated by the objects F{n) for n finite; in the same way K is the dual
of the full subcategory of MocIt^ generated by the G(n)'s. In
particular, the previous construction defines a functor H: TZ >T which is a
morphism of theories, since H{G{n)) = F{n).
Prom the equivalence cp of 3.9.1, an object M £ Modr can be identified
with the functor
ModT(-,M): T >Set, F{n) ^ ModT{F{n),M).
Its composite with H produces the functor
TZ >Set, G{n)^ModT{F{n),M).
Considering the isomorphisms
ModT(i^(n),M) ^ Modr{HG{n),M) ^ Modn{G{n),W{M))
we deduce that this last functor corresponds to W{M) € ModT^ again via
the equivalence (/? of 3.9.1. Thus composing with H indeed corresponds
to applying W. D
Corollary 3.9.3 Let f: R >T be a ring homomorphism. The functor
"restricting the scalars along f",
Modr > ModT^,
which maps a right T-module M to M provided with the scalar
multiplication
M X R ^^^ ^ >M X T >M
is algebraic, D

166 Algebraic theories
on
a
Diagram 3.15
3.10 Commutative theories
A group, written additively, is commutative when the axiom x-hy = y-\-x
holds for each pair x, y of elements. More generally an n-ary
operation a: A^ >A on a set A is commutative when, for every n-tuple
(ai,..., a^) € A^^ the value of a(ai,..., an) is unaffected by a
permutation of ai,..., a^. One could be interested in an algebraic theory T in
which every morphism a: T^ >T^ gives rise, in the models, to a
commutative operation in the previous sense. But, except in the degenerate
cases, this never happens since the projections pi'. T^ >T^ already
fail to satisfy this commutativity requirement.
So, by a commutative theory, we do not mean a theory where every
operation is commutative, but one where every operation commutes with
every operation, in particular with itself. Let us give a precise definition.
Definition 3.10.1 Let T be an algebraic theory. We call T a
commutative theory when, given morphisms a: T^ >T^ and f3: T'^ >T^,
the square in diagram 3.15 commutes.
To write the previous definition in terms of variables, as in 3.2.3, it
suffices to consider a matrix {xij)i<i<n,i<j<m of elements and the axiom
C{a{xn, • • • , Xnl), . . . , Oi{xim, • • • , Xnm))
= a(/3(a:ii,..., xim), • • •, P{xni, • • •, Xnm))'
For example, if a,/3 are two binary operations written -h and x, the
previous axiom becomes
(xii + a:2i) X (a:i2 + 0:22) = {xn x 0:12) + @:21 x 0:22),
which aheady indicates that the theory of commutative rings is definitely
not a commutative theory. Now if a and C are both the binary operation
+, the axiom becomes
(xii 4- X21) 4- (a:i2 + X22) = {xn + 0:12) + @:21 + 0:22);

3.10 Commutative theories 167
when the operation -f is associative and admits a 0 element, this reduces
to the usual commutativity law
^21 +^12 = ^12+^21.
The aim of this section is to give several characterizations of
commutative theories.
First of all, we generalize a well known definition.
Definition 3.10.2 Let T be a presentation of an algebraic theory and
A, B, C three T-models. A set theoretic mapping f: A x B >C is a
T-bihomomorphism when
Mae A B >C, b^f{a,b),
MbeB A >C, a^f{a,b),
are T-homomorphisms.
Theorem 3.10.3 Let T be an algebraic theory and 71 its corresponding
canonical presentation. The following conditions are equivalent:
A) T is a commutative theory;
B) for every operation a: T^ >T^ and every Ti-model Aj the
mapping
A^ >A, (ai,...,an) t->a(ai,...,an),
is a Ti-homomorphism;
C) for every pair A^B ofTi-models, the set ModxiC^,-B) ofTi-homo-
morphisms from AtoB is provided with the structure of a Ti-model
when we define
o^ifu • • •, fn){a) = a(/i(a),..., fn{a))
for each operation a: T^ >T^, Ti-homomorphisms fi'. A >B
and element a £ A.
Under these conditions, the construction defined in C) is part of a
symmetric monoidal closed structure on Modr (see 6.1.3). The
corresponding unit object is the free model on one generator. Moreover, given two
objects A^Be Modri ? ^^^ tensor product A® B gives rise to natural
bijections
Ti - Bihom(A X B, C) ^ Modri {A ® B, C)
where A,B^C e Modri suid Bihom indicates the set of bimorphisms.
Proof If C is an m-ary operation, the second condition means precisely
the relation
/3(a(aii,..., ani),..., a(aim,..., anm))
= a{P{an,..., aim), •.., /3{ani, • • •, anm))

168 Algebraic theories
for every matrix {aij)ij of elements in every 7i-model A. Thus it holds
when the theory T is commutative. Conversely applying this relation
to the T-model r(T^^^,-) and the projections piji T^^^ >T^ as
elements yields precisely the relation /? o a^ = ao P'^,
To prove A) => C), let a: T^ >T^ and C: T^ >T^ be
operations. Considering homomorphisms /i,..., /n- A >B and elements
a, ai,..., am € A, we define
Q^(/i, • • •, fn){a) = a(/i(a),..., /^(a))
and prove that a(/i,..., fn) is a 7i-homomorphism. Indeed, since the
/i's are T-homomorphisms and A) holds
= a(/i(/3(ai,..., ttrn)),..., fn{P{ai,..., a^))
= a(/?(/i(ai),..., /i(arn)),...,/3(/n(ai), • • •, /n(am)))
= /3(o^(/l(ai), • • • , /n(ai)), . . . , a{fi{am), . . . , /n(am)) j
= /3(o^(/l, • • • , /n)(ai), . . . , 0^(/l, . . . , fn){am))'
The pointwise definition of the operations implies immediately that all
the 7i-axioms are satisfied in Modri {A, B), since they are in B; thus the
set Modri (A, B) has been provided with the structure of a 7i-model.
Conversely, let us prove C)=>B). For any operation a: T'^ >T^ and
every 7i-model A, consider the projections pi,...,Pn- A^ >A. By
assumption the mapping
Q^(Pi,---,Pn): A"" >A, (ai,...,an) h-> a(ai,... ,an)
is a 7i-homomorphism, which is precisely the content of B).
Given a commutative theory T, the previous construction extends
immediately to a bifunctor
(ModrJ* X Modri ^Modri, (A^) ^ UodrM^B),
Indeed, given a morphism /: B >C, one defines
Modr, {A, /): Modr, (A B) ^Modr, (A C), g >/ o g.
This is a Ti-honiomorphism as given an operation a: T" >T^, TJ-

3.10 Commutative theories 169
homomorphisms 5^1,... ,5^71 • ^ ^-B and an element a e A,
f o {a{gi,... ,^n))(a) = f[a{gi{a),... ,^n(a)) j
since / is a 7i-homomorphism. Next given h: D >A, a 7i-homomor-
phism, one constructs in the same way Modri (^, B) and it is obvious we
have defined a bifunctor.
Observe now that the free 7i-algebra F(l) on one generator (see 3.8.1)
is a "unit" for this bifunctor. Indeed we have bijections, for every 7i-
model A,
ModTi (^A). ^) = Set(l, U{A)) ^ U{A).
This composite bijection maps a 7i-homomorphism /: F(l) >A to
the element /(*) G A, where * stands for the generator of F{1). The
structure of Modri (F(l), A) indicates that this is a 7i-homomorphism,
thus a 7i-isomorphism since U reflects isomorphisms (see 3.3.3). Let us
also recall that U is precisely the functor represented by that "unit"
F(l); see 3.3.3.
To construct the tensor product A<S^B of two T^-models A, B, consider
the free Ti-model F{U{A x B)) on the set U{A x B) ^ U{A) x [/(B);
see 3.8.1 and 3.4.1. We consider also the canonical morphism of the
adjunction
Vu(A)xU(By U{A) X U{B) >UF{U{A) x U{B)), (a,6) ^ (a,6),
and for simplicity write (a, 6) = ^f/(A)xf/(B)(^5^)- For every operation
a: T^ >T^ and elements ai,..., a^ G A, b £ B we consider the
elements
(a(ai,..., an), b), a((ai, 6),..., {an, b)),
where the operation a is thus evaluated in A and F(U{A) x U{B)), In
the same way, for elements a e A and 61,..., 6n G B we consider the
elements
(a, aFi,..., 6n)), a((a, 61),..., (a, 6n)).
We consider the smallest congruence on F{U{A) x U{B)) generated by
all those pairs
[{a{au ...,an),b),a((ai,6),..., (an,6))j,
((a, aFi,..., 6n)), oc((a, 61),..., (a, 6n))),

170 Algebraic theories
i.e. the intersection of all sub-7i-models of F{U{A) x U{B)) which are
equivalence relations and contain all those pairs (see 3.5.1). Write R
for this congruence. By definition A<S^ B is the coequaUzer in Modri of
the two projections ff If(TI(A) x U{B))\ write a ® 6 to denote the
equivalence class of (a, 6) in this coequaUzer (see 3.5.5). The mapping
A X B >A ®B, (a, 6) t-> a ® 6
is certainly a 7i-bihomomorphism, by definition of the congruence R.
Next if /: A x B >C is a 71-bihomomorphism, the set theoretic
mapping
U{f): U{A) X U{B) ^ U{A x B) >U{C)
correspond, by adjunction, with a 71-homomorphism
h: F{U{A) X U{B)) >C
such that, in particular, /i((a,6)) = f{a^b). Since / is a bihomomor-
phism, h identifies all the pairs of elements which generate iZ; since the
kernel pair of /i is a congruence (see 2.5.6.g), the minimaUty of R impUes
it is contained in that kernel pair. In particular h coequalizes the two
projections R ^F(U{A) x U{B)^ and thus factors uniquely through
the coequaUzer A® B via a 7i-homomorphism g. We have thus defined
a mapping
Ti-Bihom(Ax B,C) >ModTi(A® B,C), f ^ g,
where 7i-Bihom stands for the set of 7i-bihomomorphisms and the
mapping g is given by g{a ®b) = f{a, b) for every pair (a, b) e Ax B. This
last equality proves the injectivity of the mapping. The surjectivity is
obvious since, given a 7i-homomorphism g: A® B >C, the relation
/(a, b) = g{a ® b) defines a 7i-bihomomorphism f: Ax B >C which
is mapped to g by the previous construction.
Let us now observe the existence of a bijection
Ti-Bihom(A xB,C)^ Modr, {A, Modn {B, C)).
A 7i-bihomomorphism f: Ax B >C is mapped to
(p: A >ModTiE,C)
defined by
<p{a): B >C, b^ f{a,b).

3.10 Commutative theories 171
Each (p{a) is a 71-homomorphism because / is a 7i-bihomomorphism.
Moreover, given an operation a: T^ >T and elements ai,..., a^ G A
and b e B,
(p{a{ai,...,an)){b) = /(a(ai,...,an), b) = a(/(ai, 6),..., f{an, b))
= a{(p{ai){b),,,, ,(p{an){b))
= a{ip{ai),...,(p{an)){b)
again since / is a 7i-bihomomorphism; this shows that (f itself is a 71-
homomorphism. This correspondence is of course bijective; its inverse
maps a 71-homomorphism h: A >ModTi(-B,C) to
ip{h):AxB >C, (a,6) >h{a){b),
which is indeed a 71-homomorphism in the first variable, since h is, and
a 71-homomorphism in the second, since each h{a) is.
Putting together the previous results, we have got bijections
ModTi {A ® B, C) ^ ModTi (^. Modri {B, C)).
It is simple to check their naturality as well as the compatibility
conditions proving that Modr has thus been provided with the structure of a
symmetric monoidal closed category. D
The reader will have observed that the construction of the tensor
product and the proof of its factorizing property for 71-bihomomorphisms
does not require the commutativity of the theory. But the tensor
product obtained in this way is often not interesting when the theory is not
"commutative enough". For example in the case of two groups A, B one
must have
(a -h a') ® F -h b') = {{a -h a') ® 6) -h ((a -h a') ® b')
= (a ® 6) -h (a' ® 6) + (a ® b') + (a' ® 6'),
(a + aO ® F + 60 = (a 0 F + 6')) + (a' 0 F + 6'))
= (a ® 6) + (a ® b') + (a' ® 6) + (a' ® 6'),
which indicates a high degree of commutativity in A<S^ B,
Examples 3.10.4
3.10.4.a The theory of sets is certainly commutative since its only
operations are projections, which obviously satisfy condition B) of 3.10.3.
3.10.4.b The theory of abelian groups is commutative since, given two
abelian groups A, B, one gets a structure of abelian group on Ab{A, B)

172 Algebraic theories
by defining
if + 9)ia) = f{a)+g{a),
(-/)(«) = -(/(a)),
0(a) = 0.
Observe that, for example, the relations
(/ + 9){a + a') = /(a + a') + f{a + a')
= f{a) + na')+gia)+gia')
= f{a)+g{a) + f{a')+g{a')
^ if + g){a) + if + g){a')
indeed require the commutativity of B.
3.10.4.C The theory of modules over a commutative ring i2 is a
commutative theory. If /: A >B is a linear mapping and r E R^ the fact
that
rf:A >B, a^r{f{a))
is iZ-linear indeed requires the commutativity of i?; given s E R
{rf){sa) = r{f{sa)) - r(s/(a)) = s{rf{a)) = s(r/)(a).
3.10.4.d A commutative theory has at most one constant, i.e. one 0-ary
operation. Indeed given 0-ary operations a,/3: T^ >T^, the relation
ao pP = P o a^ ioT commutativity reduces to a = /3. In particular, the
theory of pointed sets (see 3.3.5.i) is commutative.
Counterexample 3.10.5
Let T be an algebraic theory. The fact that Modr accepts some structure
of a symmetric monoidal closed category does not imply a priori the
commutativity of the theory T: condition 3.10.3.C) refers indeed to
a specific such structure. And in fact this implication is not valid at
all. For a counterexample, consider a non-commutative group G and the
corresponding theory of G-sets (see 3.3.5.h); since G is not commutative,
the l-ary operations do not commute with each other and the theory is
not commutative. Nevertheless we shall see in 5.2.6, volume 3, that the
category of G-sets is cartesian closed, thus synmietric monoidal closed
in a very canonical way (see 6.1.5).

3.11 Tensor product of theories 173
3.11 Tensor product of theories
In this section, we investigate a rather useful way to construct new
theories from given ones. One first trivial way to reahze this is to proceed
by limits or colimits; this is indeed possible since we have:
Proposition 3.11.1 The category of algebraic theories and their mor-
phisms is complete and cocomplete.
Proof Let us write Th for the category of algebraic theories. For every
pair (n, m) of natural numbers, we get a functor
Mn,^:Th ^Set, T ^TiT'^.T'^),
Given any functor r: J^ >Th with J^ a small category, we define a
theory T by
In more explicit terms
T{T'',T'^) = limT(/)(T^,r^).
Since the morphisms of Th are functors, they respect composition and
identities, and the category structure on each t(/) induces a category
structure on T. We shall prove that T is the limit of r. Since the
morphisms of Th respect the projections of the products, the family of i-th
projections T^ >T^ in each t(/) produces a corresponding projection
in T, proving finally that T is an algebraic theory. Observe that the
terminal object of Th is thus the degenerate theory T where T{T'^^ T'^) is
always a singleton; the unique arrow T^ >T'^ is therefore an
isomorphism and the unique T-model is the singleton.
The construction of colimits in Th is easy as well, but more
technical. By 3.7.2, the theory of sets is an initial object in Th. Now if
F, G: 71 ^72 are two morphisms of Th, we get a presentation of a
new algebraic theory by choosing all the operations and axioms of 72,
together with the axiom
(F(a))(xi,...,Xn) = (G(a))(xi,...,Xn)
for every operation a: T^ >T^ of 71; we write T for the corresponding
algebraic theory. By 3.2.6 every operation /?: T'^ >T of 72 induces an
m-ary operation of T (see 3.2.9) and this correspondence is easily seen to
define an algebraic functor H: Ti >T. Just by the construction of T
as in 3.2.6, H is the coequalizer of F, G in Th. To construct the coproduct
T = IIz6/'^(^ ^ ^) ^^ ^'^' ^^^ takes as operations the disjoint union

174 Algebraic theories
of all the operations of all the individual theories and one keeps all the
axioms of the individual theories. Let us observe that given two indices
2, j, a projection pki T^ >T^ in % and the corresponding projection
p'j^: T^ >T^ in 7}, the two axioms
Pk\^li • • • 5 ^n) — ^ki Pkv^'^') • • • 5 ^n) — ^k
imply automatically the axiom (see 3.2.5)
Pk{Xi, . . . , Xn) = P'ki^l, . . . , Xn)
in T. This indicates that the obvious embeddings % >T are mor-
phisms of theories. Just by construction, T is the coproduct of the T^'s.
D
In fact the most useful construction on algebraic theories is the
"tensor product" of two theories, sometimes combined with a coequalizer
to construct the tensor product of two theories over a third one. To
understand the importance of these constructions let us start with an
apparently different question.
Definition 3.11.2 Let ^ be a category with finite products. If T is
an algebraic theory, a model of T in ^ is a functor F: T >^
preserving finite products, A morphism of T-models in ^ is just a natural
transformation.
We shall not develop universal algebra in a category ^ with finite
products; let us nevertheless indicate that generalizing to this context
the results of the previous sections requires rather strong assumptions
on ^. Our interest is mainly in the following theorem.
Theorem 3.11.3 Let T and TZ be algebraic theories. There exists an
algebraic theory, written T® 7?., such that the following three categories
axe equivalent:
A) the category of T-models in MocIt^;
B) the category Moclr(8>7^ ofT® K-models in Set;
C) the category ofTZ-models in Modr-
Moreover there are canonical morphisms of theories
T >T®n, n >T®n,
Proof Let us first define T (®TZ as the theory constructed by
performing the coproduct of T and TZ and imposing the requirement that every
T-operation commutes with every 7J-operation. More precisely, a
presentation of T(8OJ is obtained in the following way. One takes as operations
all the operations of T and of K; one takes as axioms all the axioms of

3.11 Tensor product of theories 175
T and of TZ; moreover, for every T-operation a: T^ >T^ and every
7?.-operation /?: K^ >R^^ one requires the axiom ao C'^ = /d o oH^ in
T®ll (see 3.10.1), i.e. the axiom
a(/3(xii, . . . , Xim), . . . , /3(Xnl, . . . , XnmS)
= /3(a(Xii, . . . , Xnl), . . . , OL{Xxrn, • • • , ^nm))
for every nxm matrix {xij)ij of variables. We recall that ifp^: T^ >T^
and p^: R^ >R^ are the i-th projections in T and TZ^ the axioms
in T and 7?. imply immediately the axiom (see 3.2.5)
p-(Xi,...,Xn) =Pi{Xi,...,Xn)
in T <S>'R and both p^ and p''^ are identified in T <S>TZ with the i-th
projection pii S'^ >S^, In particular the obvious mappings
sr: T >T ® n, sn- TZ >T ® n
are functors, because T ®Ti satisfies all the axioms of T and 7?.; and
morphisms of theories, because they respect projections. Since the
previous construction is perfectly symmetric in T and K^ it suffices now to
prove the equivalence of A) and B).
Let us work with the presentations Ti^TZi oi the theories as in 3.3.4.
The considerations of 3.3.4 can be repeated here without any change to
conclude that a 71-model in ModT^^ is an 7?.i-model M together with an
7?.i-homomorphism qm- M^ >M for each T^-operation a: T^ >T;
these 7?.i-homomorphisms qm are required to satisfy all the equalities
corresponding to the 7i-axioms. Forgetting the 7?.i-structure of M, the
mappings qm provide the set M with the structure of a T^-model. Thus
all 71- and all 7?.i-operations are realized in the set M and all 71- and all
7?.i-axioms are satisfied. To have a (T® 7?.)i-model, it remains to check
the commutativity condition corresponding to the choice of operations
a: T^ >T^ in T and C: R"^ >R^ in Tl. Given a matrix {xij)ij of
elements of M (i < n, j <m), the relation
/^V^V^ll? • • • 5 ^nl j? • • • 5 \Xim^ • • • 5 Xfim))
= {0{Xn, . . . , Xim), . . . , /3{Xnl, . . . , Xnm))
holds in the product M^ = M x ... x M (see 3.4.1), so that, since qm

176 Algebraic theories
is an 7?.i-homomorphism,
O^M (/3(^ll, • • • , Xim), . . . , /3(Xnl, • • • , Xnm))
= OLM (/3((^11, • • • , ^nl), '",{Xi 771, ... , XjifYijj J
= /3(c^m(^11, • • • , Xnl), . . . , O^M(a:im, • • • , Xnm)),
which is the required commutation property.
Conversely, assume the set M is provided with the structure of a
(T®7?.)-model; it is in particular an 7^-model, and the realization of a T-
operation a: T^ >T^ is a mapping qm- M'^ >M with the property
O^M (/3(^ll, • • • , Xim), . . . , P{Xnl, • • • , Xnm))
= P{(^m{Xii, . . . , Xnl), . . . , O^Mixim, • • • , Xnm))
for every 7^-operation /?: i?'^ >i?^ and every matrix {xij)ij of
elements of M. Because of the pointwise action of the operation /3 on the
product M^ = M X ... X M, this relation becomes
O^M i^{{Xii, . . . , Xnl), . . . , (Xim, • • • , Xnm)))
= P{(^m{Xii, . . . , Xnl), . . . , O^M(a:im, • • • , Xnm))'
But saying this for each /? is just saying that aM is an 7?.i-homomorphism
and thus M is a 7i-model in ModT^^.
The case of morphisms is easy since given two (T ® 7?.)i-models M
and N^ a. {T <S> 7?.)i-homomorphism is a mapping which commutes with
all the operations of (T ® 7?.)i, thus with all the operations of 71 and
all operations of TZi. This is the same as giving an 7?.i-homomorphism
which commutes with all the operations of T. D
Proposition 3.11.4 The tensor product of theories as defined in 3.11.3
is associative, commutative and admits the theory of sets as a unit.
Proof We have already observed that the construction of T ® 7?.
in 3.11.3 is completely symmetric in T and K^ from which follows the
commutativity of the tensor product.
Let us now compute the tensor product of a theory T with the theory
of sets. The only operations of the theory of sets are the projections, thus
(T(8M)i has the same operations as T. If a: T^ >T^ is a T-operation
and pki T^ >T^ is a projection, the commutativity requirement
becomes
a{xik, .. . , Xnk) = Oi{xik, • • • , Xnk)

3.11 Tensor product of theories
n
F/ \^^
177
11®T'
V
-^11®^;T
Diagram 3.16
which is a tautology. Thus T ® 5 is just T.
For the associativity condition, consider three theories T, 7?., V. Both
theories T®G?.®V) and (T®7?.)®V admit a presentation given by all the
operations of T, Ti, and V together with the axioms of these theories and
the requirement that every operation of a theory commutes with every
operation of one of the other two theories. This proves the associativity.
D
A useful generalization of the notion of tensor product given in 3.11.3
is expressed in the following definition.
Definition 3.11.5 Let T, 7?., V be algebraic theories and F: V >TZ,
G: V >T morphisms of theories. By the tensor product 'R®\;T we
mean the theory obtained as the coequahzer in diagram 3.16. In other
words, it is the theory obtained from TZ<S>T by adding the axiom
(F(a))(xi,...,Xn) = (G(a))(xi,...,Xn)
for every operation a: V^ >V^ ofV.
In order to study some examples, it is useful to note the following
lemma.
Lemma 3.11.6 Let G be a set provided with two binary additions -f, ©
with the same zero element 0. Suppose the two additions commute, i.e.
(a -f 6) e (c -h d) = (a e c) -h F e d)
for elements a, b^c^d E G. In these conditions the two additions coincide,
are associative and commutative.
Proof Choosing 6 = 0 = c yields a^d = a-^-d. Putting 6 = 0 impUes
the associativity and putting a = d = 0 gives the commutativity. D

178 Algebraic theories
Examples 3.11.7
3.11.7.a A group G in the category of groups is just an abelian group.
Indeed let us consider a group (G, -f, —, 0) provided with a group
structure (G, 0, ©, 0) in the category Gr of groups. The terminal object of
Gr is the zero group, so that the 0-ary operation T^ >T^ of the
theory of groups is reaUzed by the zero homomorphism @) >G; in other
words, the two zero elements 0 and © of G coincide. Next 0: G^ >G
is a group homomorphism for the first structure, yielding
(a -h 6) 0 (c -h d) = 0(a -h 6, c -h d) = 0((a, c) + F, d))
= 0(a, c) -h 0F, d) = (a 0 c) -h F 0 d).
Applying lemma 3.11.6, we conclude that both group structures coincide
and are commutative.
3.11.7.b Let R^T be two rings with units. A left i?-module in the
category of right T-modules is thus a set M provided with the structure
of both a left i?-module and a right T-module, the operations of the left
i?-module being T-linear. As in 3.11.7.(a), lemma 3.11.6 implies that
both additions coincide. It remains to express the statement that left
multiplying by r G i2 is right T-linear, which means r{mt) = {rm)t
for elements m G M^ t £ T. We have just described the notion of a
iZ-T-bimodule.
Now write K for the theory of left iZ-modules and T for the theory
of right T-modules. We shall observe that K <S> T is just the theory
of (J? ® T)-modules. Recall that the tensor product i? ® T (as abeUan
groups) is indeed a ring, with multiplication given by
We must prove the coincidence between the notions of R® T-module
and J?-T-bimodule. If M is an {R ® T)-module, we provide it with both
an R- and a T-multiplication via the formulae
rm = (r ® l)m,
mt = A ®t)m^
where r G J?, m G M, t G T. It is obvious that
r{mt) = (r (g) 1)A (g) t)m = (r (g) t)m = A (g) t)(r (g) l)m = (rm)t.
Conversely if M is an iJ-T-bimodule it suffices to define

3.12 A glance at Morita theory 179
to get the structure of a (i? ® T)-module. The remaining verifications
are straightforward.
3.11.7.C Analogously to the previous example, consider now two ho-
momorphisms /: V >i2, g: V >T of commutative rings with units.
Write K, T,V for the corresponding theories of modules and
F:V >n, G:V >T
for the corresponding morphisms of theories (see 3.12). One deduces
immediately from 3.11.7.b that Ti,®^T is the theory of (i?(8)vT)-modules.
3.12 A glance at Morita theory
The Morita problem for algebraic theories is simple to state: find
conditions on two theories 7?., T, so that the corresponding categories ModT^
and Modr of models are equivalent. The difficulty of the problem lies in
the fact that a such non-obvious equivalence can never been realized via
an algebraic functor. Indeed we have
Proposition 3.12.1 An algebraic functor W\ Mod^? >Mod7^
between two algebraic categories is an equivalence of categories if and only
if the corresponding morphism of theories is an isomorphism.
Proof Write H: K >T for the corresponding morphism of
theories. If H is an isomorphism, W is an equivalence. Conversely consider
the situation and the notation of 3.9.2, where it is proved that H maps
the free 7^-model G{n) on n generators to the free T-model F{n) on
n generators, which is the reflection of G{n) along W. But if W is an
equivalence, it has a left adjoint which is an equivalence and the
restriction H of this left adjoint to the corresponding full subcategories is also
an equivalence. But since by definition a morphism H of theories is bi-
jective on the objects, it is an isomorphism as long as it is an equivalence
(see 3.4.3, volume 1). D
The classical Morita theorem is concerned with the case where both
theories T, TZ are theories of modules on a ring. We assume some
familiarity with classical module theory.
Theorem 3.12.2 Let R,T be two rings with unit. The following
conditions are equivalent:
A) the categories Mod^ Sind Modr of right module are equivalent;

180 Algebraic theories
B) there exist an R-T-bimodule P and a T-R-bimodule Q such that
the following isomorphisms of bimodules hold:
P®tQ = R. Q®rP^T.
Two rings of this kind are called "Morita equivalent^\
Proof Suppose we are given an equivalence cp: ModiR >ModT. The
functor cp has an adjoint (see 3.4.3, volume 1), thus it preserves the
zero morphisms and all finite limits and colimits (see 3.2.2, volume 1).
In particular (p preserves the addition of morphisms (see 9.6.2). The
structure of left i?-module of R given by the multiplication
R X R >R, (r, r') ^ rr'
can equivalently be given by the group homomorphism
R >UoAr{R, R), r^{r' ^ rr').
Since the functor (/? is an additive equivalence of categories, the abelian
group Modi?(i2, R) is isomorphic to MoAt{^{R)^ ^{R)) so that we get a
group homomorphism
R >ModT{(p{R),(p{R))
thus a structure of left i?-module on (p{R) such that left multiplying by
r € J? is right T-linear. In other words, we have provided ^{R) with the
structure of an iZ-T-bimodule (see 3.11.7.b). We denote this bimodule
by P.
The functor
—<S^rP: Modi? >ModT
admits a right adjoint, namely
ModT(-P, —): ModT ^^Mod^,
thus —®rP preserves colimits. But the equivalence of categories (p
admits also a right adjoint (see 3.4.3, volume 1), thus preserves coUmits
(see 3.2.2, volume 1). Those two functors coincide on the free i?-module
on one generator F(l) ^ R (see 3.8.4) just because R®rP ^ P ^ cp{R).
By preservation of colimits they coincide on each F{X) = IIa;€X-^({^})'
thus finally on each M € Modj?; see 3.8.9. Observing that a right linear
mapping /: R >R is just left multiplying by r = /(I), one concludes
inmiediately that (p{f) = /(8)hP, from which by the previous colimit
argument (p and —(8)hP coincide on the morphisms as well.

3,12 A glance at Morita theory 181
In a completely analogous way one proves that "the" inverse
equivalence ip: ModT >ModR has the form —®tQ for some T-i?-bimodule
Q. The relations ^ o (/? = id, ipoip = \d applied respectively to R and T
yield the required isomorphisms P®tQ — R^ Q®rP — T. The converse
implication is obvious. D
Corollary 3.12.3 Let R be a commutative ring with a unit. Then R is
Morita equivalent to the ring R^^'^ of n x n matrices.
Proof Now R^ is an R-R^^'^-modnle: R acts componentwise on R^
while R^^'^ acts by right matrix multiplication, viewing an element of
R^ as a. line matrix. In the same way R^ is an R^^'^-R-modnle: R acts
componentwise on R^ while R^^'^ acts by left matrix multiplication,
viewing an element of iZ"^ as a column matrix. And it is well known that
By 3.12.2, R and R'''''' are thus Morita equivalent. D
To avoid the impression that it is quite common to have Morita
equivalent rings, let us study the case of commutative rings.
Lemma 3.12.4 Two rings with units which are Morita equivalent have
isomorphic centres.
Proof We use the notation of 3.12.2. The conditions on P, Q imply
immediately that
Q<S)R-: R-Mod >T-Mod, P®r-: T-Mod >R-Mod
are also equivalences between the corresponding categories of left
modules. In particular
T-Mod(T, T) ^ R-Mod{P, P)
while in 3.12.2 we had found the isomorphism
ModR{R,R)^ModT{P,P).
In this last case, left multiplying s e Rhy the element r e Ris applying
r(-): R >R, while left multiplying p G P by r G iZ is by definition
applying ^{r{')). Since cp is an equivalence of categories, the last
isomorphism thus restricts to an isomorphism
R-ModR{R,R) ^ R'ModriP^P)

182 Algebraic theories
between the subgroups of left iZ-linear mappings. An analogous
argument in the first case yields
T-ModT(T,T) ^ i?-ModT(P,P).
A right iZ-linear mapping /: R >R is left multiplying by /(I), while
/ is left linear when it coincides with right multiplying by /(I). Thus
/ is both left and right iZ-linear when r/(l) = /(l)r for every r e R^
hence when /(I) belongs to the centre Z{R) of R. The previous two
isomorphisms thus imply that when R^T are Morita equivalent, their
centres are isomorphic. D
Corollary 3.12.5 Two commutative rings with units are Morita
equivalent if and only if they are isomorphic. D
Counterexample 3.12.6
If iZ is a ring with a unit and TZ is the corresponding theory of right
modules, R is also the free module on one generator (see 3.8.14). The
monoid TZ{R^^R^) of l-ary operations is thus the dual of the monoid
Modi^(i?, R) (see 3.8.5). But the right linear mappings R >R are just
the mappings
f: R >i2, X H^ rx
for every r £ R. Given another element s £ R
f(^{x)) = f{sx) = rsx.
Hence TZ{R^, R^) is just the ring R with the opposite multiplication as
composition of the arrows.
Now consider a commutative ring with unit R and the ring R^^'^ of
n X n matrices (n > 2). Write TZ for the theory of iZ-modules and T
for the theory of iZ'^^'^-modules. The monoid TZ{R^,R^) is commutative
since R is, but the monoid T{T^,T^) is not commutative because R^^'^
is not. Thus the theory TZ is certainly not isomorphic to the theory T.
Nevertheless the corresponding categories of models are equivalent, as
attested by corollary 3.12.3.
3.13 Exercises
3.13.1 Prove that the category of fields is not algebraic. [Hint: there is
no terminal obj^t.]

S.13 Exercises 183
3.13.2 The category of small categories is not algebraic. [Hint: it is not
regular, see 2.4.6.]
3.13.3 The category of posets is not algebraic. [Hint: it is not regular;
consider the quotient of {a < 6, c < d} identifying 6, c and pull it back
along the inclusion of {[a] < [d\}.]
3.13.4 The category of complete lattices is not algebraic. [Hint: filtered
unions do not commute with finite intersections; consider the colimit of
the intervals {0,..., n} of N, which is NU {oo}, and pull it back over the
singleton {oo}.]
3.13.5 Let T be an algebraic theory; write U: Modr >Set for the
corresponding forgetful functor, F: Set >Modx for the free model
functor and rj: Ig^^ => UF for the canonical natural transformation of the
adjunction. Prove that
A) F is faithful iff each rjx is injective,
B) some rjx is not injective iff the axiom x = y holds in T for two
distinct variables x, y.
3.13.6 Prove that given an algebraic theory T, the free model on one
generator is a regular generator for Modr-
3.13.7 Given an algebraic theory T, prove that the forgetful functor
U: Modr >Set does not preserve epimorphisms. [Hint: consider the
theory of commutative rings with a unit; see 1.8.5.f, volume 1.]
3.13.8 If iZ is a commutative ring with unit, prove that the theory of R-
modules is a category which admits both a covariant and a contravariant
isomorphism with its dual.
3.13.9 In an algebraic category, prove that a free model P does not in
general satisfy the condition: given an epimorphism p: X »Y and a
morphism f: P >Y, f factors through p. Compare with 4.6.1,
volume 1 and 3.8.6, this volume. [Hint: see 3.13.7.]
3.13.10 Prove that Z is not a dense generator in the category of abelian
groups (compare with 3.8.10).
3.13.11 A functor G: 3f >Set is dominated by a family {Xi)i^i of
objects of 3f when, for every Y e 3C and every y G G{Y), there exist
iel.x e G{Xi) and /: Xi >Y such that y = G(/)(x).
A) Prove that a functor G: Set ^Set which preserves filtered colimits
is dominated by the finite sets.

184 Algebraic theories
B) Prove that the previous implication is not an equivalence. [Hint:
consider the identity on the finite sets and extend it by the value
{*} on the infinite sets.]
C) Replace condition B)(e) of 3.9.1 by "t/F is dominated by the finite
sets" and prove the theorem still holds.
3.13.12 Let M be the ring of infinite real matrices with the property
that each Une has only a finite number of non-zero elements; addition is
pointwise and multiplication is the matrix product. Prove that if n^m
are positive integers, the free M-module on n generators is isomorphic
to the free M-module on m generators. [Hint: given two matrices A, B^
construct another whose lines are successively the first line of A, the first
line of B, the second line of A, the second line of B, ]
3.13.13 Prove that tensoring with T over R produces the left adjoint
to the algebraic functor of 3.9.3.
3.13.14 Let T be an algebraic theory and consider the forgetful functor
U: Modq >Set. Prove that T is isomorphic to the category whose
objects are the functors U'^{n G N) and whose morphisms are the natural
transformations between those functors.
3.13.15 We consider two universes U eV and write Set for the category
of W-sets. We consider the category 3C whose objects are the pairs (^, U)
where
A) ^ is a category with ^ G V and ^(X, Y) eU for X, F G ^,
B) U: ^ >Set is a functor such that the natural endo-transforma-
tions U => U constitute a U-set.
The morphisms W: (^, U) > {9, V) are just the functors W: ^ >Q)
such that V o W = J7. If Th is the category of algebraic theories in the
universe ZY, we get a contravariant functor Th >9C mapping a theory
to the corresponding category of models provided with its forgetful
functor and a morphism of theories on the corresponding algebraic functor.
Prove that this functor has an adjoint mapping the pair (^, t/) to the
full subcategory of Fun(^, Set) generated by the functors JJ'^{n G N).
(This is the so-called "structure semantics" adjunction.) By 3.13.4, the
functor Th >dC is full and faithful.
3.13.16 Let iZ, T be two rings with units and corresponding theories
7J, T of right modules. Consider an iZ-T-bimodule M viewed as a
finite product preserving functor F\ VP ^Modr; see 3.8.15 and 3.11.7.
Prove that the functor —(8)hM: Mod^e >Modr is the left Kan
extension of F along the Yoneda embedding TZ^ «^-* Mod^e.

3.13 Exercises 185
3.13.17 Consider an algebraic theory T and a full subcategory ^ of
Modr- Prove that the following conditions are equivalent:
A) ^ is an algebraic category and the inclusion ^ C Mod^ is an
algebraic functor;
B) ^ is closed in Modr under arbitrary products and the inclusion
creates monomorphisms and strong epimorphisms (i.e. if /: A >B
is a monomorphism in Modr and B e^, then A G ^; if /: A >B
is a regular epimorphism in Modr and A G ^, then B e^).
[Hint: given a T-model M, consider all the regular epimorphisms of
the form M »C with C e ^] compute the corresponding product
JJC indexed by all those morphisms and let t{M) be the image of the
induced factorization M > H^'; prove that r yields the left adjoint
to the inclusion; apply 3.13.11.]
3.13.18 Given a commutative ring R with a unit, consider a retract
r, s: S ^R^, r o s = Is, oi some finite power of R in Mod/?. When S
is a generator of Mod/?, prove that R is Morita equivalent to the ring
Modi?(S', S) of S'-Unear endomorphisms of S. This generalizes 3.12.3.

4
Monads
The classical definition of a monoid is to give a set M together with a
binary operation
MxM >M, {x,y)\-^xy
which is associative and admits a unit element written 1. As a
consequence, with every finite sequence (xi,..., Xn) of elements of M is
associated a composite element xi... Xn defined inductively in the usual
way:
• with the empty sequence is associated the element 1;
• with the sequence (xi,..., Xn) is associated the element xxn, where
X is the element associated with the sub-sequence (xi,... ,Xn-i).
In other words, a monoid can also be seen as a set where "every
finite sequence has been given a composite". Let us make this a precise
definition.
Given a set M, we write T{M) for the set of finite sequences of
elements of M. Given a mapping /: M >N, T{f): T{M) >T{N) is
the mapping sending the sequence (xi,..., Xn) to the sequence of
corresponding images (/(xi),..., /(xn)).
We want to provide the set M with the structure of a monoid via a
mapping
^:T{M) >M
which associates with every finite sequence (xi,... ,Xn) of elements of
M a new element £(xi,... ,Xn) € M which we call the "composite of
the sequence". We must investigate the axioms requiring £ to be exactly
the composition law induced by a monoid structure on M.
186

Monads
187
M ^^ >T{M)
TT{M) ^^ )T{M)
T{M) —^ M
Diagram 4.1
First of all the "normalization condition" ^(x) = x must certainly be
satisfied. We shall express it by considering the mapping
>T{M), X i-> (x)
and requiring as an axiom the commutativity of the triangle part in
diagram 4.1.
Next a general associativity condition of the type
e(e(a},...,aij,...,e(ar,...,a:r„))
= ^(a},...,ai^,a?,...,a-J_\,ar,...,a-^)
must be satisfied as well. Observe that choosing a finite sequence of
finite sequences of elements of M is the same as choosing an element of
rr(M).
The previous axiom thus involves the concatenation mapping
IXM-.TTiM) >T{M),
{ial...,al),...,{aT,...,a^J)
I—> (tti, • • • ,tt^i,tti,... ,a^^_^,ai ,...,a^^j,
which constructs a "composite sequence" from a sequence of sequences.
The general associativity axiom can be expressed by the commutativity
of the square in diagram 4.1.
It is now easy to verify that the previous definition is equivalent to
the classical definition of a monoid. It suffices to define a multiplication
on M via
M xM-
->M, {x,y)^^{x,y).
The associativity rule is just
x{yz) = ^{x,^{y,z)) = ^{^{x),^{y,z)) =^{x,y,z).

188 Monads
A'M
1t(m)
T{fiM)
fJ'M
T{M)
TT{M)-fj^T{M)
Diagram 4.2
Now put 1 = ^( ), the composite of the empty sequence.
ix = i{)m = i{i{U{^))=i{ x)
= ^{x) = X
and in the same way x = xl.
Thus M has been provided with the structure of a monoid and one
proves inductively that ^ is just the composition for this monoid
structure:
• ^( ) is indeed the element 1;
• if ^(xi,..., Xn-i) = Xl... Xn-i, then
^(Xi, . . . , Xn) = ^(^(Xi, . . . , Xn-l), ^{Xn))
^ SV*^1 • • • *^n—1? Xn) ^ yXi . . . Xn—xjXn
^=- X\ . . . Xji—\Xji.
Before leaving this example, one should observe that the two mappings
^MyfJ^M which have been used to describe the two axioms themselves
satisfy interesting relations, namely the commutativity of all pieces of
diagram 4.2. The commutativity of the triangles means that the
concatenation of both sequences
((a:i,...,Xn)), ((xi),...,(a:n))
yields the sequence (xi,... ,a:n); the commutativity of the square
expresses the associativity of the concatenation process.
4.1 Monads and their algebras
With the previous example in mind, we define:

4-1 Monads and their algebras
189
/i* It
TTT-^ ^ TT
It */i
Diagram 4.3
>r(C) TT{C)-
no-
Diagram 4.4
TT
M
^T{C)
-> C
M
-> T
Definition 4.1.1 A monad on a category ^ is a triple (T, £:,/i) where
T: ^ >^ is a functor and e: l<g => T, /i: TT => T are natural
transformations satisfying the commutativity conditions
/i o (g: * It) = It = fio (It * £:), /i o (/i * It) = fio (It * /i)
(see diagram 4.3).
Definition 4.1.2 Let T(r, £:, /i) be a monad on a category ^. By an
algebra on this monad is meant a pair (C, ^) where C G ^, ^: T{C) >C
and ^oec = Ic? ^oT{^) = ^ofic; see diagram 4.4. If (D,Q is another
T-algebra, a morphism f: (C,^ >{D, Q of T-algebras is a morphism
f: C >D of^ such that fo^ = CoT{f); see diagram 4.5.
Proposition 4.1.3 Let T = (T, £:, /i) be a monad on a category ^. The
T-algebras and their morphisms constitute a category, written ^^. D
The category ^^ is also called the "Eilenberg-Moore" category of the
monad.
Proposition 4.1.4 Let T = {T,e,fji) be a monad on a category ^.
Consider the forgetful functor
U:<^'^ >^, (C,^)h^C, f^f
A) U is faithful;
B) U reflects isomorphisnis;

190 Monads
T{C)-—^^T{D)
» D
f
Diagram 4.5
C) U has a left adjoint.
The left adjoint FofU maps C e^ to {T{C), fic) and f: C >C' to
T{f) : (r(C),/ic) >{T{C'),fic')' The unit of the adjunction is just
e: leg => UF = T and the counit rj: FU >lcg^ is given hy rj^co = C-
Proof The faithfulness is obvious. Next, if /: (C,^) >{D^Q is such
that / is an isomorphism in ^,
f o^oTif-') = coTif) oT{r') = c = f o r' oc.
from which Cor(/-i) = /"^oC and/-^r (D,C) )► (C, 0 is a morphism
of T-algebras, inverse to /.
Now if C G ^, let us first observe that the axioms for being a monad
imply immediately that (T{C)^^c) is a T-algebra. We shall prove that
{T{C),fic)^ together with the morphism ec'- C >T{C), constitutes
the reflection of C along [/; see 3.1.1, volume 1. Given a T-algebra (D, Q
and a morphism /: C >D in ^, we must prove the existence of a
unique g: [T{C),fic) >{D,0 in ^^ such that g o sc = f- For the
uniqueness, it suffices to observe that, with the conditions imposed on
g = goficoT{ec) = Cor(^) oT{ec) = CoT{f).
For the existence it suffices to check that g = (^ o T{f) satisfies the
required conditions. It is indeed a morphism of T-algebras, since
po/zc = CoT(/)o/ic7 = Co/iDorr(/) = Cor(c)orr(/) = Cor(^),
and it is such that goec = f-, because
goec = C,oT{f) oec = C^eo o f = f.
Observe that given a morphism h: C >C' in ^, putting (D, C) =
{T{C')^Ijlc') and / = ec o /i in the previous construction yields the

4-1 Monads and their algebras 191
relation g = fic o T{ec') o T{h) = T{h)\ thus by 3.1.3, volume 1,
T{h): (r(C),/ic) >(r(cO,/icO
is the value of the left adjoint F of [/ on h: C >C'. By construction,
UoF = T,
By 3.1.5, volume 1, the unit of the adjunction is just e. If ry is the
counit of the adjunction, ^(c,^) is obtained by putting / = Ic in the
previous construction (see 3.1.5, volume 1), yielding ^(c,^) = C- D
Definition 4.1.5 Let T = (!',£:,/i) be a monad on a category ^.
Consider the forgetful functor U=^'^ > ^ and its left adjoint F: ^ > ^^,
mapping C to {T{C)^fic)' A T-aigebra is free when it is isomorphic to
one of the form F{C) = (T{C)^fic), for some object C G ^.
Proposition 4.1.6 Let T = (T, £:, /i) be a monad on a category ^. The
full subcategory of^^ generated by the free T-algebras is equivalent to
the following category ^j:
• the objects of^j are those of^;
• a morphism f: C >D in ^j is a morphism f: C >T{D) in ^;
• the composite of two morphisms f: A >B, g: B >C in ^j is
given in ^ by the composite
A—^—>T{B) ^^^^ )rr(C)—^^^-^r(C);
• the identity on an object C of^j is just ec- C >T{C) in ^.
Proof Choosing another morphism h: C >D in ^t, the associativity
of the composition is attested by the following relations in ^:
= fiD° fiT{D) ° TT{h) o T{g) o /
= HDO Tub o TT(h) o T{g) o /
= liDoT{iiDoT{h)og)of-
while the identity axioms, for a morphism /: C >D in <^t, mean in
AtD o T{f) oec = liDO £t(d) ° f = f, Md o T{eD) o / = /•
Writing ^j for the full subcategory of 'i^ generated by the free T-
algebras, we get a functor y>: '^j >^j just by putting
• ^(C) = (r(C),/xc)forCG<^T,
• vif) = HDO T{f) for /: C >T{D) in <<r, i.e. /: C >D in 'i^j.

192 Monads
Observe that (p{f): (T(C)^^c) >{T{D)^^d) is indeed a morphism
in ^T since
liD O Tifif) =flDO T{flD) O TT{f) =fjiDO fJiT(D) O TT{f)
= llDO T{f) o /ic = (p{f) o lie.
Now (/? is a functor: indeed it preserves identities since
ip{ec) = fjico T{ec) = 1t(C)
and also composition since, with the notation of the statement,
^{9) o ^U) = Mc o r(^) o fiB o T{f) = fic o fir(C) o TT{g) o T{f)
= /ic o T{fic) o rr(^) o T{f) = fic o T{fic o T{g) o f) = ^{g o /).
By the choice of ^t, every object of ^j is isomorphic to an object
of the form (pC; thus it remains to prove that (p is full and faithful (see
3.4.3, volume 1). This is essentially 4.1.4C). If/, g: C ^)T{D) are such
that fjLD o T{f) = fiDO T{g),
f = fiDO£T{D)of = fiDoT{f)oec = fiDoT{g)oec = fJ'DoeT(D)^9 = 9,
so that (p is faithful. The functor (p is also full since, given
h: (r(C),//c) >{T{D),fiD)
in ,^Tj the composite hoec'- C >T{D) is such that
fiD oT(hoec) = fiDOT(h) oT{ec) = hoficoT{ec) = h. D
The category ^j is also called the "Kleisli" category of the monad T.
Using only proposition 4.1.6, one has a full and faithful functor
^T >^\ CH^(r(c),/ic),
identifying, up to equivalence, ^j with the full subcategory of ^"^
generated by the free T-algebras.
Corollary 4.1.7 Let T = {T,e,fi) be a monad on a category ^. The
functor
*T >^, CH^r(C), {f:C-^D)^fjiDoT{f)
is faithfal, reSects isomorphisms and has a left adjoint given by
Proof Via the equivalence of 4.1.6, this is just the situation of
proposition 4.1.4 restricted to the full subcategory of free T-algebras. D

4.2 Monads and adjunctions 193
Diagram 4.6
4.2 Monads and adjunctions
Given a monad (T^e^fi) on a category ^, we have produced two adjoint
pairs: the Eilenberg-Moore adjunction and the Kleisli adjunction. Let
us fix notation. We have already written
[/:(^T ^^^ p.c^ ^c^T^ p^^
for the first adjoint pair; one of the canonical natural transformations
of the adjunction (see 3.1.5, volume 1) is just e: 1<^ ^ U o F = T (see
4.1.4); we shall write tj: FoU => Icg^ for the other one. Let us also write
F:^T >^, G:^ >^t, GHF
for the Kleisli adjoint pair (see 4.1.7); the first canonical natural
transformation of this adjunction is again e: 1<^ =>V oG — T^ while we write
7: GoV =^ 1<^TP for the other one.
The object of this section is to prove that every adjunction induces
a monad and to compare the original adjunction with the Eilenberg-
Moore and Kleisli adjunctions of the monad.
Proposition 4.2.1 Let R: dC >^ and L: ^ >3f^ constitute an
adjoint pair, with L left adjoint to R; write a: 1<^ => i? o L, /3: L o i? => 1^
for the canonical natural transformations of this adjunction. Under those
conditions, putting
T — RoL, e = a, /jl = 1r^ P^Il
yields a monad (T, e^^) on ^. Moreover there exist functors J: ^j > 3^
and K: SC ^^'^ such that the isomorphisms RoK '^V, UoJ ^ R
hold (see diagram 4.6). Moreover J o K is isomorphic to the canonical
inclusion of^j in ^^ (see 4.1.6), J is full and K is full and faithful.
Proof The statement defines T: ^ >^, e: leg =^ T and fi: ToT =^ T.
Let us check the three conditions for a monad (see 4.1.1), which we
deduce from the triangular identities of the adjunction (see 3.1.5, volume 1)

194
Monads
LRLR{X
Plr(x
'LR{X)
LRiM
f3.
X
LR{X)
/3x
Diagram 4.7
> X
and the naturality of /3; this naturality implies the commutativity of the
square in diagram 4.7 for every object X ^ SC. Next
/i o (e * It) = (li? * /? * 1l) o (a * 1h * 1l)
= ((li? * /3) o (a * 1h)) * 1l = li? * 1l
= 1t,
/i o (It * e) = (li? * /3 * 1l) o (li? * 1l * c^)
= li? * ((/? * 1l) o Al * a;)) = li? * 1l
= 1t,
/i O (/i * It) = (li? * /? * 1l) O A^ * /? * Ijr, * 1^ * Ijr^)
= li? * (/? o (/? * 1l * li?)) * 1l = li? * (/? o Al * li? * /?)) * 1l
= (li? * /? * 1l) O A^ * Ijr^ * 1^ * /? * Ijr^)
= /io (It */i).
We are now able to define K and J.
if:^^ ,^, if(C)-L(C), K{f) = f3LDoL{f),
where /: C )►!) in ^t is thus an arrow /: C >RL{D) in ^; K{f)
is thus the composite
L{C)—^^^i—^LRL{D) ^^^^^ >L{D).
Observe that iiT is a functor since, given g: D >E in ^j (see 4.1.6),
K{g) o K{f) = I3l(e) o L{g) o 0ld o L{f)
= Pl{e) o Plrl(E) o LRL{g) o L{f)
= Pl(e) o LR{PLiE)) o LRL{g) o L{f)
= Cl{e) oL{fiEO T{g) o /) = /3^(^) o L{g o f)
= K{gofl

4:2 Monads and adjunctions 195
K{1d) = Pl{D) o H^d) = Pl{d) ° L{sd) = Pl{d) o Hao)
Moreover, given an arrow /: C >D in ^j,
RK{C) = RL{C) = T{C) = ViC),
RK{f) = R{I3l(d) o Hf)) = R{PLiD)) o RL{f) = fiD o T(/) = V{f),
which proves the relation RK = V.
Next let us construct J;
J: ^ ><^'^, X ^ {R{X),R{f3x)), x ^ R{x),
where x: X > F is a morphism of ^. Observe first that
R/3x: TR{X)^—=RLR{X) >R{X)
provides R{X) with the structure of a T-algebra.
-R(/?x) o £fl(x) = R{l^x) o aR(x)
R{^X) O /ifl(X) = Ril^x) O R{f3LR(X)) = R{Px O fSlRiX))
= R{^x o LR{l3x))
= R{Px)oT{R{l3x)).
On the other hand R{x): {R{X), R{/3x)) > {R{Y),R{I3y)) is a
morphism of T-algebras since
R{Py) o TR{x) = R{Py) o RLR{x)
= R{i3y o LR{x)) = R{x o Px)
= R{x)oR{l3x).
J is obviously a functor, since R is, and by definition U o J = R and
J o K = (f^ where ip: ^j >^'^ is the functor defined in 4.1.6.
Since JoK is full and faithful, K is faithful and J is full. But given a
morphism h: L{C) >L{D) in ^, h corresponds by adjunction with a
morphism /: C >RL{D) = T{D) in ^, i.e. a morphism /: C >D
in ^T such that K{f) -= h. So K is full as well. D
To complete the statement of proposition 4.2.1, let us consider the
cases where the original adjunction is already the Eilenberg-Moore or
the Kleisli adjunction of a monad.

196 Monads
Proposition 4.2.2 Let T = (T^e^fi) be a monad on a category ^.
Via the construction of 4.2.1 ^ both the Eilenberg-Moore and the Kleisli
adjunctions of T generate the monad T.
Proof We keep the notation described at the beginning of this section.
By construction, we have UF = T with e: 1<^ => UF as the unit of
the adjunction. It remains to prove that ^ — U ^r}^ F^ where ry is the
counit of the adjunction. But given a T-algebra (-^,0? '^{x,0 = ^ by
4.1.4, thus for C G ^, U{rjF(c)) = U{ri^T(C),fj,c)) ^ ^c- The case of the
Kleisli adjunction follows immediately from the Eilenberg-Moore case
and proposition 4.1.6. D
Combining propositions 4.1.1 and 4.2.2 we find that every adjunction
generates a monad and, among all the adjunctions which generate the
same monad, the Kleisli adjunction and the Eilenberg-Moore adjunction
are somehow "extremal", in the precise sense explained in 4.2.1.
Now let us emphasize a special case of interest.
Proposition 4.2.3 Let T = (T, e, /i) be a monad on a category ^. With
the notation of the beginning of this section^ the following conditions are
equivalent:
A) the forgetful functor U: ^^ >^ is full and faithful;
B) the counit rj: FoU => Icgi of the adjunction F -iU is an isomorphism;
C) the multiplication fi: T oT ^T of the monad is an isomorphism;
D) for every T-algebra (X,^), ^: T{X) >X is an isomorphism in ^.
Such a monad is called an idempotent monad.
Proof A) and B) are equivalent by 3.4.1, volume 1. B) implies C)
since we have observed in proving 4.2.2 that /i = [/ * ry * F. And D)
implies C) since (r(X),/ix) is a T-algebra for every X G ^; see 4.1.5.
Going back to 3.1.5, volume 1, we know that given a T-algebra (X, ^),
the morphism rfix,^)' FU{X,^) >{X,^) is the unique one with the
property t/(r7(x,o) ^ ^f/(x,o = lf/(x,o; so ri(x,0' ^(^) ^^ i^ the
unique morphism yielding ^ o T{ri(x,o) == ^(^,4) ° ^x and r)(x,o ^^x =
Ix' But ^ is such a morphism, thus ri(^x,^) = ^- This proves the
equivalence of B) and D).
It remains to prove, say, that C) implies D). Prom fix o ^t(X) =
1t(x) = Mx o^(^x) we deduce that eT(x) = ^(^x) is an isomorphism.
Prom ^ oex = Ix we deduce T(^) o T{€x) = 1t(X), thus T(^) = fix is
an isomorphism as well. Prom T(^) o eT{x) = ^x o C we get that ex o^
is an isomorphism, thus in particular ^ is a monomorphism. But since

4-3 Limits and colimits in categories of algebras 197
^ o £:x = Ix, this implies that ^ is an isomorphism with inverse £x] see
1.7.5, volume 1. D
Corollary 4.2.4 Let ^ be a category There is a coincidence, up to
equivalences of categories, between
A) the reflective subcategories of^ (see 3.5.2, volume 1),
B) the categories of T-algebras for the idempotent monads T on ^.
Proof Consider a reflective subcategory i: ® ^^ ^ with corresponding
reflection r: ^ >S>; write, as in 4.2.1, a: 1*^ => i o r and /3: r o i => 1^
for the canonical natural transformations of the adjunction. By 3.4.1,
volume 1, /3 is an isomorphism. If T = (r,£:,/i) is the corresponding
monad as in 4.2.1, then /i is an isomorphism as well since /i = 1^ o/3o 1^.
Thus the monad T is idempotent. Moreover the comparison functor
J: S> >^'^ described in 4.2.1 is now an equivalence of categories.
Indeed for every T-algebra (C, ^), the isomorphism ^: ir{C) = T{C) >C
indicates that C is in the replete subcategory ®; on the other hand
giving a morphism x: X >Y in S> is equivalent to giving a morphism
x: {X, Px) > {Y, Py) in ^^, just by naturality of C. Thus every
reflective subcategory of ^ is equivalent to the Eilenberg-Moore category of
the idempotent monad it generates.
On the other hand, applying 4.2.3 and 3.4.1, volume 1, we know
already that an idempotent monad T on ^ admits as category of T-
algebras, up to equivalence, a reflective subcategory of ^. D
4.3 Limits and colimits in categories of algebras
Proposition 4.3.1 Let T = (T^e^fi) be a monad on a category ^. Let
G: 2 >^^ be a functor such that UG: 9 >^ has a limit; then G
has a limit which is preserved by U: ^^ >^. In particular if^ admits
some type of limits, ^^ admits the same type of limits and they are
preserved by U.
Proof Let G: 3) ^^"^ be a functor such that [/ o G: Q) ^^ has
a limit; we must prove that G\ Q) >^^ has a limit, which will of
course be preserved by JJ since U has a left adjoint (see 4.1.4 and 3.2.2,
volume 1).
Write \pu\ L >UG{D))j^^^ for the limit of UG: 3) >^, where
the forgetful functor U: ^^ >^ has been defined in 4.1.4. For each

198 Monads
Z? G ®, let us write G{D) = (C/G(D),^d)- The family of morphisms
T{L) ^^^^^ ) TUG{D) —^^—>UG{D)
is a cone on UG since, given d: D >D^ in ®,
C/G(d) o ^^ o T{pd) = ^D' o TUG{d) o T{pd) = ^d' o T{pd').
Therefore we get a unique factorization ^: T{L) >L such that po^^ =
^D o T{pd) for each Z) G ®.
It is now routine to check that {pn'- {L,^) >{UG{D)^^d)) is the
limit of G. First,
Pdo^oel^^do T(pd) oel^^do euG(D) opD= Pd,
from which ^oel ^'^l-i^Y definition of a limit. In the same way
PDoiopL^ino T{pd) opL = ^DO PuG(D) o TT{pd)
- iD o niD) o TT{pd) - ^D o T{pn) o r@
= PDO^or@,
from which we obtain ^o pi = C ^ ^@- Thus (L,^) is already a T-
algebra and the relation ^d o T(pd) — Pd^ i means precisely that the
arrow po'- {L,^) >{UG{D),^d) is a morphism of T-algebras. Since
U: ^^ >^ is faithful (see 4.1.4), these morphisms constitute a cone
on the functor G.
Now given another cone {cid'- {M,Q >{UG{D),£^d)) j^^^ on G, we
get in ^ a unique factorization m: M >L such that for each D G ®,
PDom = q^At remains to prove that m: (M, Q > (L, ^) is a morphism
in ^^. This follows immediately from the relations
PDomoC = qDO(^ = ^DoT{qD)o^D^T{pD)oT{m) =pDO^oT{m). D
Proposition 4.3.2 Let T = (T, £:, p) be a monad on a category ^.
Let G: ^ )^^'^ be a functor such that UG: Q) ^►^ has a coiimit
preserved by T and by T o T; then G has a cohmit which is preserved
by U: ^^ >^. In particular if^ has some type of colimits preserved
by T, then ^^ has the same type of colimits and these are preserved by
U: <ifT ycg^
Proof Let G: ^ >^'^ be a functor such that U o G: ^ >^ has a
colimit preserved by T and T o T; we must prove that G: ^ >^'^ has
a colimit preserved by U.
Write (sd: UG{D) >L)j^^^ for the colimit oiUG: 3f >^, where
U: ^ >^ is the forgetful functor of 4.1.4. To fix the notation, or

4-3 Limits and colimits in categories of algebras 199
each D e 9, let us write G{D) = {UG{D),^d)' Observe that the
morphisms ^d'- TUG{D) >UG{D) are the components of a
natural transformation TUG => UG^ just because for every d: D >D^
in ®, G{d) is a morphism in ^^. Therefore we get a unique
factorization ^: colimTUG > colim C/G, i.e. a unique morphism ^: r(L) >L
such that ^ o T{sd) = sd^^d for each D G Si.
It is now routine to check that
' De^
is the colimit of G. First of all,
^0£lOSd = ^^ T{sd) O euG(D) = SdO^dO euG(D) = ^D,
from which ^o£:j[^ = lj[;^, by definition of a colimit. Next, since the colimit
of TTUG is just (rr(L), TT{sd))j,^^.
^OflLO TT{sd) =^0 T{sd) O fjiuG(D) =Sd0^dO flUG(D)
= SdO^DO f(^)D = ^ O T{sd) O T{^d)
= ^oT{OoTT{sn)
implies ^o fij^ = ^or(^), from which (L,^) is already a T-algebra. The
relation ^o T{sd) = sd^^d implies that sd- {UG{D)^^d) >{L^ ^) is
a morphism of T-algebras.
Now given another cocone {ro'- {UG{D)^^£)) ^{^X))d^^ on G,
we get in ^ a unique factorization m: L >M such that for each D G
®, m o 5£) = r£). It remains to prove that m: (L, ^) >{MX) is a
morphism in ^'^. Since (r(L),rE£)))^^^ is the colimit of TG, this
follows immediately from the relations
CoT{m)oT{sD) = C'^Tiro) = vdo^d = mosDO^D = mo^oT{sD)^ □
Ij^mina 4.3.3 Let T = (T, £:, /i) be a monad on a category ^. For every
T-aJ^febra (G, ^), the following diagram is a coequaJizer in ^^:
(rr(G), /iT(c)) ==t (r(^)' MC7) —^-^ (G, 0.
This coequalizer is transformed by the forgetful functor U: ^^ >^ in
the absolute coequalizer of diagram 4,8,
Proof Now {TT{C),fiT{C)) and (r(G),/ic) are T-algebras (see 4.1.4
and 4.1.5); fic is a morphism in ^"^ by definition of a monad (see 4.1.1);
T(^) is a morphism in ^^ by naturality of /i and ^ is a morphism in ^^

200 Monads
^T(C) EC
TTiC^ \T(C\ ^ > C
no
Diagram 4.8
by definition of a T-algebra (see 4.1.2). Moreover in ^, the consideration
of the morphisms
TT{C)< ^^^^^ T{C)^-^ C
yields ^o fic = ^oT{^) by definition of a T-algebra; fic oSt(C) = 1t(C)
by definition of a monad (see 4.1.1); ^ o g:^ = 1G by definition of a T-
algebra (see 4.1.2) and r(^) o £t(C) = ^c o ^ by naturality of e. Thus in
<if, U{^) is the absolute coequalizer of (/ic',r(^)); see 2.10.2, volume 1,
and diagram 4.8.
It remains to prove that we also have a coequalizer in ^^. We already
have ^ofic = ^oT{^) and if /: (r(C),/ic) >{D,Q is a morphism in
^"^ such that f o fic = f o r(^), the composite
C—^^-^r(C) ^—>D
is the unique factorization of / through ^ in ^; see 2.10.2, volume 1.
It suffices to prove that / o eq: (C,^) >{D^Q is a morphism of T-
algebras. Indeed
C o T{f) o T{ec) = / o /ic o T{ec) = f
= fofico eT(C) = / o r@ o £t{C)
= f 0£C0^. D
It can be useful to observe that in the previous lemma, the morphism
T{ec)' T{C) >TT{C) is in fact a common section for fic and r(^),
/xc o T{ec) = 1t(C), r@ o T{ec) = 1t(C),
by definition of a monad (see 4.1.1) and a T-algebra (see 4.1.2).
Proposition 4.3.4 Let T = (T^e^fj) be a monad on a cocomplete
category %, The following conditions are equivalent:
A) the category ^'^ ofT-algebras has coequalizers;
B) the category ^'^ ofT-algebras is cocomplete.

4^3 Limits and colimits in categories of algebras 201
M]jQ°^(«)
Timi)
(t(UQ),/^Uc.)
ic,0
Diagram 4.9
Proof By 2.8.1, volume 1, it suffices to prove that if ^^ has coequalizers,
then ^^ has also coproducts. The proof is inspired by the statement of
lemma 4.3.3.
Consider a family (Q, ^i)iei of objects of ^^ and compute its coprod-
uct {si-. Ci ^LIi€/<^i)ie/ in <^. Write (a^: T{Ci) >]Xi^iT{Ci))^^j
for the corresponding coproduct in ^ of the images along T. The
morphisms T{si) yield a unique ^-morphism 5: LIi6/^(^*) ^^ (Uie/^O
such that so cFi= T{si) for each i G /.
Let us now define the coproduct (C,^) = LIi6/(^*'^*) ^^ ^^ "^^^ ^'^^
coequalizer ^^ given by diagram 4.9 where, for clarity, we have
omitted writing "i G /" all the time. The two first objects are just free T-
algebras (see 4.1.4 and 4.1.5) and to give perfect sense to the definition
of (C,^), it remains to prove that the two left arrows are morphisms in
^^. The arrow T (fj ^i) is a morphism in ^^ just by naturality of /i while
MuCi o ^E) is a morphism in ^'^ because
/iuQ o T (/iucj o TT{s) = flue, o /iT(UQ) o TT{s)
= mCi O ^E) O fiuT(Ci)'
Thus (C, ^) has been correctly defined.
For each index i € / T{si): (r(Q),/icO ^ (^dJ^i) .MucJ is a
morphism of T-algebras, by naturality of /i. Because of lemma 4.3.3, the
relations
q o T{si) ofic,=qo /iuc^ o TT{si) =qo /iuc^ o T{s) o T{ai)
^Q^T{]l^i) oT{ai) = q o T{si) o TC^i)

202 Monads
imply the existence of a unique morphism S^: (Ci,^i) >{C,^) of T-
algebras such that SiO^^ = qoT{si). This yields the canonical morphisms
of the coproduct.
Now let us choose a family of morphisms U: {Ci^ ^i) > (D, Q in ^^\
we must prove the existence of a unique t: (C, ^) >{D^Q such that
toYti =ti for each index i G /. In ^, the morphisms t^ yield a unique
It: JJCi >D such that tx o 5^ = t^. Prom this we get, by 4.1.4,
morphisms of T-algebras
(t (II q) , line) ^^^ (T(Z>), mo) ^ {D, C).
On the other hand the morphisms T{ti) of ^ yield a unique factorization
v: Y[T{Ci) >T{D) such that voai = T{ti). And from the relations
'^ ° (II ^v ° ^i = '^ ° ^i ° ^i ^ ^i ° ^i ^ C o r(ti) = c o 1^ o ^i
one deduces that t^ o (fj^i) = (^ ov. In an analogous way the relations
C o T{u) o 5 o cTi = C o T{u) o T{si) = C o T{ti) = Covoai
imply C o r(tx) 05 = ^0'^- We are now ready to prove that ^ o r('^)
coequalizes fiud ^T{s) and TdJ^i):
C o T{u) o /iuQ o T{s) = c o /iD o rr(tx) o rE)
= C o r(C) o TT{u) o T{s) = C o r(C) o r(i;)
= Cor(^)or(U^,).
By definition of (C,^), we get t: (C,^) >(D, C), a unique morphism
of T-algebras such that toq = (^o T{u). This implies
toEi0^i = toqoT{si) = CoT{u)oT{si) = CoT{ti)=Uo^i,
from which t o S^ = t^ since ^^ is an epimorphism {^i oe^ = 1^.).
It remains to prove the uniqueness of t. Let r: (C, ^) >- (D, Q be
such that r o Si = ti for every index i e I. Observe first that
roqosocFi = roqoT{si) = roSiO^^ = tiO^i = uoSiO^i = uo (JJ^) ^<^i'
This implies r oqo s = uo (W^^i) and thus
Toq = roqoT (jj^^) o T (U^c,)
= roqo fjLuCi o T{s) o T (jJ^Ci)
^ro^oT{q)oT{s)oT(jlec,)

4.3 Limits and colimits in categories of algebras 203
^CoT{r)oT{q)oT{s)oT(jlsc,)
= CoT{u)oT(jl^i) oT(jlec,)
= CoT(n),
from which r = t. D
Theorem 4.3.5 Let ^ be a complete and cocomplete regular category in
which every regular epimorphism has a section. Under these conditions,
for every monad T = (T, £:, /i) on ^:
A) the category ^^ ofT-algebras is complete, cocomplete and regular;
B) the forgetful functor U: ^^ >^ is exact;
C) the forgetful functor U: ^^ >^ preserves and reflects regular epi-
morphisms;
D) a pair of morphisms {u,v): (G^^) 1{L>,0 is a kernel pair in ^^
if and only if (tx, v) is a kernel pair in ^;
E) ^^ is exact as long as ^ is exact.
In particular, assuming the axiom of choice, all those properties are
satished when ^ is the category Set of sets.
Proof Applying 6.5.4, volume 1, a regular epimorphism p: A >B
with section 5: B >A is the coequalizer of the pair E op, 1^). Since
^(A, A) is a set, there is just - up to isomorphism - a set of regular
quotients of A, corresponding to the idempotents e G ^(A, A)\ see 6.5.4,
volume 1. On the other hand, every functor F with domain ^ preserves
regular epimorphisms, since F(p) has a section F{s) and 6.5.4, volume 1,
again leads us to conclude that F(p) is a coequalizer.
Now ^^ is complete by 4.3.1; to show it is cocomplete, we need to prove
the existence of coequalizers (see 4.3.4). Given f^g: (C,^)ZZlt(-D,C) ^^
^^, we consider all the morphisms h: (D,C) ^(-^?x) ^^ ^^ with the
following properties:
A) hof = hog;
B) h: D >X is a regular epimorphism in ^.
We know there is just such a set of morphisms h so that we can compute
in ^'^ the product of all the T-algebras (X, x), for all the possible h. The
family of morphisms h induces a factorization k through the product
and we consider its image factorization fc = iop in ^ as in diagram 4.10.
We shall prove that / is in fact a sub-T-algebra of the product HaiC-^' x)
and p is the coequaUzer of /, g.

204
Monads
t D
X ^
Ph
Diagram 4.10
-* /
-n.^
T(i)
T{D) ^^^ »T{I) ^—^> J >—^—>T{P)
D
a
•• d
P - I
Diagram 4.11
-> P
Let us write (P,7r) = Hhi^.x) for the product in ^'^. The
factorization k: (I?,C) >{P^'^) is a morphism of T-algebras, thus the outer
part of diagram 4.11 in ^ is commutative, where we know that T{p) is
a regular epimorphism. The arrow T{i) has no reason to be a mono-
morphism, but nevertheless by 4.4.5, volimie 1, we get a factorization
a: J >I making the diagram commutative. Putting a = a o g, we
shall prove that (/,a) is a T-algebra; by definition of a, this will make
p: (i?,C) >{I,(^) and i: (/,a) >(i^,7r) morphisms of T-algebras.
Let us recall that p, T{p) and TT{p) are (regular) epimorphisms. By
naturality of e, /i,
aoei op = ao T{p) o sd = P^ C ^ ^d = P^
ao fijo TT{p) = ao T{p) o/i£,=po(;o/i£,=po(;o T{Q
= aoT{p) oT(C) = aoT{a) oTT{p),
from which a o e/ = 1/ and a o /ij = a o T{a),
Thus we have produced p: (D, C) > (/, ot) in ^^. Prom the relations
Phoiopof = p^okof = hof=ihog=phokog=pf^oiopog
we deduce i o p o f = i o p o g and thus p o f = p o p, since z is a

4-3 Limits and colimits in categories of algebras 205
monomorphism. Moreover, if the morphism h: (D,C) ^i^^x) ^^ ^^
is such that hof = hog^ by construction we have Ph^i: (/, a) > (X, x)
in ^^ such that {phoi)op = h. This factorization is unique since p is an
epimorphism in ^. Thus p: (D, C) ^(^?<^) is the coequalizer of {f^g)
in ^^ and ^^ is cocomplete.
The functor U: ^^ >^ preserves limits (see 4.3.1) but also regular
epimorphisms, since the construction of the coequalizer p = Coker (/, g)
in the first part of the proof indicates in particular that p is a regular
epimorphism in ^. By 2.5.7, volume 1, U preserves coequalizers of kernel
pairs since it preserves kernel pairs and regular epimorphisms. But since
U reflects isomorphisms (see 4.1.4), U reflects coequalizers of kernel pairs
(see 2.9.7, volume 1) and thus reflects regular epimorphisms.
As U: ^^ >^ preserves and reflects puUbacks and regular
epimorphisms, the puUback of a regular epimorphism in ^^ along any morphism
is again a regular epimorphism, since this is the case in ^. Thus ^^ is
regular (see 2.1.1).
Let us now consider an exact sequence
Zd—^l^q
in ^ (see 2.3.1) and a section 5: Q >D of the regular epimorphism
q. Since q o 1^ = g o 5 o g, we get a unique factorization t: D >C
through the kernel pair (tx,v) such that Id = uot^ soq = vot. In other
words every exact sequence in ^ is an absolute coequalizer (see 2.10.2,
volume 1).
Next, if tx, v: (C, ^) > (D, Q is a pair of morphisms in ^^ such that
u^v: C ^D is a kernel pair in ^, with coequalizer q: D >Q^ we
have just observed that this coequalizer is preserved by every functor,
thus in particular by T and ToT. Therefore 4.3.2 applies, showing that
Q can be provided with the structure (Q, a) of a T-algebra in such a way
that q: (D,^ >{Q^a) is the cokernel of (u^v) in ^^. Since U reflects
kernel pairs (see 4.3.1), (u^v) is the kernel pair of q in ^^.
Finally ii u^v = (J?, p) ZZI^ (C, ^) is an equivalence relation in ^^,
then u^v: is an equivalence relation in ^ since U preserves
Umits (see 2.5.4). If ^ is exact, (tx, v) is a kernel pair in ^ (see 2.5.3) and
thus (it, v) is also a kernel pair in ^'^. D
Another interesting example of a category ^ satisfying the
requirements of 4.3.5 is a Grothendieck topos ^ satisfying the axiom of choice
(see 3.4.3 and 7.5.1, volimie 3).

206 Monads
Here is a final example where the cocompleteness of a category of T-
algebras can be proved; this example will take its full meaning in the
context of locally presentable categories (see 5.5.8, volume 1).
Proposition 4.3.6 Let ^ be a complete and cocomplete category and
T = (r,£:,/i) a monad on ^. If the functor T: ^ >^ preserves Hi-
filtered colimitSj for some regular cardinal k, the category ^^ of T-
algebras is gomplete and cocomplete.
Proof The completeness is attested by 4.3.1. To prove the
cocompleteness it suffices, by 4.3.4, to prove the existence of coequalizers. If
jT is the category • >• used to describe coequalizers (see 2.6.7.b,
voliune 1) we must prove that the functor
A: ^^ ^Fun(jr,^^),
mapping (A,^) to the constant functor on (A,^), has a left adjoint
(see 3.2.3, volume 1). But we know already that ^^ is complete and,
moreover, that A preserves limits since these are computed pointwise in
[jT,^"^]; see 2.15.1, volunie 1. By the adjoint functor theorem, it remains
to prove the solution set condition (see 3.3.3, volume 1). Giving a
functor F: Jf >^'^ means giving a pair f^g: (A, a)ZZI^(J5,/3) of arrows
in ^'^. A set of objects (D^, 6i) G ^^, i G /, is a solution set for F when
for each morphism h: (B,/3) >(C',7) in ^^ such that ho f = ho g^
h factors through some (Di^Si) in ^^. (Indeed, giving a cocone on F is
equivalent to giving such a h; see 2.6.7.b, volume 1.) We shall construct
such a solution set reduced to a single object of ^^. This will be done
via a transfinite induction.
For every cardinal A, we shall define objects P\,Qx and arrows pA,
9Aj i*A, va as in diagram 4.12. The diagram will be commutative with
Qx^ f = Qx^ 9, Px^ T{f) =pxo T{g). Given cardinals A < z/, there will
be transition morphisms px,u: Px >Pi^, Qxy- Qx ^Qv still making
diagram 4.12 commutative, with qx^iy o qx = g^,, px,iy o Pa = Pu and
Qi^u = 1q,, Piyiy = Ip,. We shall prove that T{Q^) ^ P^ and {(Q«,ia«)}
is the required solution set.
For the initial step of the induction, we just define qo = Coker (/, g)
and po = Coker {T{f),T{g)). Prom the relations
qooPo T{f) = qoofoa = qoogoa = qoopo T{g)
we deduce the existence of a unique uq such that uoopQ = qQO /3. Prom
T{qo) o T{f) = T{qo) o T{g) we deduce the existence of a unique Vq such
that vo opo = r(go)-

4.3 Limits and colimits in categories of algebras
207
T{qx)y
T{f)
T{A) ]T(B) —^
v\
^ Px^^^ P.
a
T{9)
f
/?
"A
Uv
t B
J^^ Q, -lh:lL^ Q^
Diagram 4.12
Now suppose the construction is achieved for every ordinal A < z/; we
shall extend it to the level u -\-l. We define Uj,^i as the coequalizer
t{p,):=zzz=iXt{Q.) -—""^^
TK)
^Qu
+1
and simply put
and necessarily
Qu-\-l = Qv.u+l O qv Piy-\-l = Piy,iy-\-l ^ Pw
One has the required commutativities since
= Ujy^i O fjLQ^ O T{V^) O Ep^ = 1X^+1 O IjLq^ O Et(Q^) O V^
and
Now suppose z/ is a limit ordinal and the construction is realized for every
ordinal A < z/. We define Qj, as the colimit of the diagram constituted
of the various Qx, A < z/, and transition morphisms between them; g^,!/
is the canonical morphism to the colimit and
Qv = 90,1/ O go = 9A,i/ O gO,A O go = Q\,iy O QX-

208 Monads
In the same way Pjy is defined as the colimit of the diagram constituted
of the various P\, X < u and transition morphisms between them; pA,i/
is the canonical morphism of the colimit and
Piy = P0,i/ O PO = PA,i/ O PO,A opo= px^u O PX-
The commutativity properties of the morphisms ux^ X < u^ imply that
the morphisms qx.i^^ux constitute a cocone on the diagram defining P^^;
thus we get a unique factorization Ui^ such that u^ ^P\v = Qx,iy ^'^x- In
the same way the various vx^ A < z/, induce a cocone T{qx^iy)ovx, A < z/,
from which we get the required morphism Vj,.
The previous construction has defined P^^+i = T{Qi,) for every
ordinal jy. Observe also that given ordinals A < z/, an analogous formula
Pa+i,i/+i = T{qx^v) holds for the transition morphisms. Indeed, by the
previous construction and the commutativity of diagram 4.12,
PA+1,i/+1 =Pv,v+l oPA+1,1/ = Vj, opxj^i^i,
= r(gA+i,.) o vx^i = T{qx+i,,) o r(gA,A+i)
= T{qx,^).
Next we note that if z/ is a limit ordinal, the successor ordinals A+1 < z/
constitute a cofinal diagram among all the ordinals A < z/ (see 2.11.2,
volume 1); therefore computing the colimit on A < z/ is equivalent to
computing a colimit on A + 1 < z/. But since T{Qj,) = P^^i and
T{qx,u) = Pa+i,i/+i for every ordinal A < /i, the previous remark
immediately implies the relation
P, ^ colimA<«r(QA),
where the regular cardinal k> has been identified with the smallest ordinal
of that cardinality. By regularity of «:, the ordinals strictly less than k>
constitute a /^-filtered diagram from which, by our assumption on the
functor T,
P« ^ colimA<«r(QA) = r(colimA<«gA) = T{Q^).
Since colimits are defined up to isomorphism, we can choose the colimits
in such a way that this isomorphism (which is v^) is the identity.
The previous isomorphism allows us to consider the pair (Q«,tx«),
where Uk,: T{Q^) >Qk' We prove it is a T-algebra. Let us consider
the coequalizer diagram defining Ujy^i. The naturality of /i implies the
commutativity of diagram 4.13, for every z/ < /^, thus the fact that
fjLQ^ is the colimit of the various /jlq^. Computing the colimit of the

4.3 Limits and colimits in categories of algebras 209
TT{Q,)-i^^T{Q,)
TT{q,,^)
T{qu,.)
TT{Q,)-j^T{Q,)
Diagram 4.13
various coequalizer diagrams and transition morphisms, for the successor
ordinals z/ + 1 < /^, we get
TK)
since T preserves the /^-filtered colimits involved and v,^ is the identity.
Thus Uk, o fiQ^ =z Uk, o T{uk,)^ which is one of the axioms for being a
T-algebra (see 4.1.2). For the other axiom observe that by definition, for
every ordinal A the relation gA,A+i = '^A+i o ^Qx holds. As for /i, the
naturality of e implies that £q^ is the colimit of the various £q^ , A < «:.
On the other hand the colimit of the transition morphisms gA,A+i is just
the identity on Q^^ since computing the colimit on A < «: is equivalent
to computing it on A + 1 < «:. Finally the definition of gA,A+i yields, at
the colimit, 1q^ = u^o eq^^ which is the required axiom.
Next we prove that g«: (J5,/3) >'(Q«j'^«) is a morphism of T-alge-
bras. By commutativity of diagram 4.12, we know already that u,^ op^ =
Qk, o C and it remains to prove that p^, = Ti^q,^). This is an immediate
consequence of the relation v\opx = T{qx)^ since computing the colimit
for A < «: yields v^op^ = T{q^) and the isomorphism v^, has been chosen
to be the identity (see 4.1.2).
Finally let us choose a morphism h: {B,/3) ^(^,7) of T-algebras,
such that ho f = hog. We just need to find a morphism of T-algebras
l^' {Qk^'^k) ^(C',7) such that k o q^ = h. We construct k by trans-
finite induction. More precisely we shall construct, for every ordinal A,
a morphism kx: Qx >C in ^ such that diagram 4.14 is commutative
and kiy = kxo qx^u for every ordinal A < z/. We shall prove that k = k^,
is the required factorization.
To initialize the process, we choose ko: Qo >C as the unique
factorization of h through the coequalizer go of (/, g) in ^. One has imme-

210
Monads
7
kx
Diagram 4.14
-> c
T{K)
Kif.
+ 1
ncy
-> c
Diagram 4.15
diately, since /i is a morphism of T-algebras,
J oT{ko) o vo o Po = J oT{ko) oT{qo) = J oT{h)
= h o C = ko o qq o C = ko o uo o Po,
from which 7 o T{ko) = ko o uo since po is an epimorphism (see 2.4.3,
volume 1).
Now suppose the construction is achieved for every ordinal A < z/; we
shall extend it to the level z/ + 1. The relations
7 o T{k^) o fjLQ^ o T{v^) = 7 o /ic7 o TT{k^) o T{v^)
= joT{^)oTT{k,)oT{v,)
= 70 T{k^) o TM
imply that the composite ^oT{kjy) factors through the coequalizer Uj,-\-i
of the pair (/ig^ o T{vi,)^T{ui,)), yielding a morphism fci^+i such that
diagram 4.15 commutes. The stated pentagonal relation is satisfied since
7 o r(fc^+i) o Vr^^i = 70 r(fc^+i) o r(g^,^+i)
= 70 r(fc^+i) o r(tx^+i) o r(^Q J
= 7oTG)oTr(Mor(^Qj
= 7 o rG) o T{ec) o T{k,) = 7 o T{k,)

4-3 Limits and colimits in categories of algebras
211
^^^T(Q«)
T(A:«)
¥T{C)
7
-> C
Diagram 4.16
On the other hand the construction is compatible with the transition
morphisms since
= 70T{K) osQ^ ^joecoK
Next suppose the construction has been realized for every ordinal
A < z/, with u a limit ordinal. Because of the compatibility conditions
with the transition morphisms g^,!/) the morphisms k\: Q\ >C define
a cocone on the diagram defining Q^^ as a colimit; therefore we get a
factorization fc^,: Qi, >C such that ki^oqx^i, = kx for each X < u. This
already takes care of the compatibility with the transition morphisms. To
prove the pentagonal condition, observe that Pjy itself has been defined as
a colimit, so that it suffices to prove the commutativity after composition
with each pa,i/j A < z/.
7 o r(fc^) o 1;^ o px^r, =70 r(fc^) o T{qx,jy) o 1;;^ = 7 o T{kx) o vx
= kxoux = kjyO qx^j, o ux
Considering now the morphism k^,, we have
Koq^ = kK,o go,« oqQ = kooqQ = h
and so just need to prove that k^,: {Qk-, u^) > (C, 7) is a morphism of
T-algebras. But again since T preserves /^-filtered colimits, computing
the colimit for A < /^ of the pentagonal diagrams, together with the
connecting morphisms, yields a commutative pentagon as in diagram 4.16.
Since by choice of the colimits, v,^ is just the identity, this proves that
k,^ is indeed a morphism in ^^. D

212 Monads
C I D ^—^ Q
V
Diagram 4.17
4.4 Characterization of monadic categories
We want now to characterize those categories which can be described as
categories of algebras for some monad on a base category.
Definition 4.4.1 A functor R: 3C >^ is monadic when there exist a
monad T = (T, £:, /i) on ^ and an equivalence of categories J: 3C >^'^
such that U o J is isomorphic to R, where U: ^^ >^ is the forgetful
functor of 4.1.4.
Definition 4.4.2 By a split coequalizer in a category ^ we mean a
situation as in diagram 4.17 where qou = qov, qos = 1q, uor = Id,
V or = s o q.
The terminology in 4.4.2 is justified by the fact that q is then the co-
equalizer of u^ V and, moreover, this coequalizer is absolute, i.e. preserved
by every functor defined on ^; see 2.10.2, volume 1.
It is useful to recall here part of the statement of lemma 4.3.3:
Lemma 4.4.3 Let T = (T, £:, /i) be a monad on a category ^. For every
T-algebra (X,^), diagram 4.8 is a split coequalizer in ^. D
Theorem 4.4.4 Let R: ^ "f^ be a functor. The following conditions
axe equivalent:
A) R is monadic;
B) (a) R has a left adjoint L;
(b) R reflects isomorphisms;
(c) if a pair u,v: X ]¥ in ^ is such that [R{u),R{v)) has a
split coequalizer in ^, then (tx, v) has a coequalizer in 3C which
is preserved by R.
Proof To prove A) => B), it suffices to show that given a monad
T = (T, e, /x) on ^, the forgetful functor U: ^^ ^^ satisfies B)(a), (b),
(c). Conditions B) (a) and B)(b) are already attested by 4.1.4. So let us
choose It, v, (C, ^) ) (£>, C) in ^ and morphisms q, r, 5 in ^ producing
a spHt coequalizer as in 4.4.2. This split coequalizer is preserved by

4-4 Characterization of monadic categories 213
RLRL{C) \ RL{C) ^ > C
RLiO
Diagram 4.18
every functor (see 2.10.2, volume 1), thus in particular by T and T oT.
Therefore 4.3.2 applies, proving that the coequalizer of (tx, v) exists in
^^ and is preserved by U.
Conversely, we use the notation of section 4.2; it suffices to prove
that the comparison functor J: ^ >^'^ is an equivalence of categories,
where T is the monad generated by the adjunction L -^ R. Since the
composite J o K is just the inclusion ^j C <^'^, we know already that J
is full. On the other hand given a T-algebra (C,^, we consider the pair
/3l(C), ^@- LRL{C)ZZXL{C) in ^ which, by lemma 4.4.3, satisfies the
conditions of B)(c); (see diagram 4.18), since T = RoL, e = a and
/i = R * C * L] see 4.2.1. Applying our assumption B)(c), we get a
coequalizer in ^,
LRLjC) IZtL(C) 2—^Q^
HO
such that, up to isomorphism, R{Q) = C and R{q) = ^. In particular
R{q) has a section ac, thus LR{q) has a section as well and is therefore
an epimorphism (see 1.7.4). By definition of q and naturality of /3,
q o LR{q) =qo L{^) = qo Plr{Q) = Cq o LR{q),
from which q = I3q, since LR{q) is an epimorphism. Finally, we have
obtained
j{Q) = (i?(g),i?(/3Q)) ^ (c,i?(g)) = (CO.
It remains to prove that J is faithful. Since J7 o J ^ i?, it suffices to
prove that R is faithful. Choose x,y: X ^Y in ^ such that R{x) =
R{y). By naturality of /3,
xopx=pYO LR{x) = /3yo LR{y) = yopx.
so that it suffices to prove that Px is an epimorphism. For this consider

214
Monads
OLRLR{X) OLR{X)
f
^r
^
R{0LR(X)) • - ^/o X
RLRLRiX) — i RLR{X) ^^^' ) R{X)
RLRi^x)
Diagram 4.19
PlR{X)
LRLR{X)^:^LR{X)
LR{l3x)
Diagram 4.20
-^ Q
X
the diagram
Plr(x)
LRLR(X) > LR{X) -
LR{f3x)
i3x
^X
in ^, whose image under i? is a split coequalizer in ^ (see diagram 4.19),
just by the triangular identities of the adjunction L -\ R (see 3.1.5,
volume 1) and the naturality of a. By our assumption B)(c), we get
diagram 4.20 in ^ where q is the coequalizer of {fiLR{X)',LR{Px))
and z is the unique factorization yielding /3x = z o q; moreover B)(c)
asserts that R{q) is just R{/3x), up to isomorphism. In particular R{z)
an isomorphism and thus z is one too, by our assumption B)(b). Thus
Px is an epimorphism since q is, and the proof is complete. D
It is natural to wonder if the composite of two monadic functors is
again monadic. This is not the case in general, as attested by coim-
terexample 4.6.4. Observe that in 4.4.4, conditions B) (a) and B)(b) are
obviously stable under composition, but condition B)(c) is not.
In the case of monadic categories over Set, we get a more remarkable
result.
Theorem 4.4.5 Let ^ be a category. The following conditions are
equivalent:
(l) there exists a monadic functor U: ^ >^Set;

4-4 Characterization of monadic categories 215
B) (a) ^ has finite limits;
(b) ^ is exact;
(c) ^ has a regular generator P;
(d) P is projective;
(e) the copower Ux^ exists for every set X.
Proof Given a monadic functor U: ^ ^-Set and its left adjoint
F: Set >^ (see 4.4.4), put P = F{1), where 1 is the singleton. Prom
the bijections, for every C G ^,
C/(C)^Set(l,C/(C)) ^^(FA),C),
we deduce that U is the functor represented by P. Now ^ has finite limits
by 4.3.1, ^ is exact by 4.3.5 and P is a generator since U is faithful (see
4.1.4, this volume, and 4.5.9, volume 1). The copowers Ux^ ^^^^* since
^ is cocomplete (see 4.3.5).
Now consider the monad T = (T, £:, /i) generating ^. Given a T-algebra
(X, ^) and an element x G X, we view x as a mapping x: 1 > U{X, ^)
and we write x: P(l) >{X,^) for the corresponding morphism of T-
algebras obtained by adjunction. Diagram 4.21 is commutative by 3.1.5,
volume 1, since ^: FU{X, ^) > (X, ^) is the component at (X, ^) of the
canonical natural transformation FU => id<^ of the adjunction F -\ U
(see 3.1.5, volume 1, and 4.1.4, this volume). But since X = Wx^ ^^d
F preserves copowers (see 3.2.2, volume 1), we have just observed, since
X ^ C/(X,0 = ^(P, (^,0), that
e. P(X) ^ T T p > (X, 0
is the morphism which, composed with the injection 5^ of the coproduct,
reproduces x. The object P is a regular generator when this morphism
is a regular epimorphism (see 4.5.3, volume 1),which is the case by 4.3.3.
It remains to prove that P is projective; by regularity, strong and
regular epimorphisms coincide (see 2.1.4). We know that U preserves regular

216
Monads
P = F(l)
k-'
-» B
ay
u{Ay
u{f)
^>U{B)
Diagram 4.22
epimorphisms (see 4.3.5). Given a regular epimorphism /: A >B in
^ and a morphism g: P >B (see diagram 4.22), g corresponds by
adjunction with an element b G U{B), which has the form U{f){a) for
a G U{A)^ since U{f) is surjective. The arrow h is the morphism
corresponding with a by adjunction and from U{f){a) = 6, we get f oh = g.
Conversely, suppose the conditions B)(a), (b), (c), (d), (e) are
satisfied. We shall apply 4.4.4 to prove that the representable functor
C/ = ^(P,-):
->Set
is monadic.
First of all, U has a left adjoint. The reflection of a set X along U is
just (Ux^'^^) where
ax:X-
.|7(U,P)^^(P,Ux^)
maps the element x G X to the corresponding canonical morphism
5a;: P ^Ux^ ^f ^^^ copower. Indeed, given C G ^ and a mapping
/: X ^«'(P,C), the family of morphisms f{x): P >C, for x G X,
yields a imique factorization g: Ux^ ^^ ^^^^ ^^^^ 9^^x = fip^)-, i-^-
such that ^{P,g) o ax = /; see 3.1.1, volume 1. We write F for the left
adjoint of U. By construction, P = F{1).
Since P is a generator U is faithful, (see 4.5.9, volume 1). In
particular U reflects monomorphisms and epimorphisms (see 1.7.7, volume 1).
Let us prove that U reflects regular epimorhisms. Given /: A >B
in ^ such that U{f) is surjective, consider diagram 4.23 where pb is
the unique morphism such that pB o s-^ = 6, for every 6 G ^{P^B).
By assumption, ps is a regular epimorphism (see 4.5.3, volume 1). But
every 6 G «'(FA), B) = Set(l,t7(jB)) corresponds then to an element
b € U{B), which can be written as U{f){a) for some a G U{A), since
U{f) is surjective. The element a, viewed as a mapping a: 1-^—>U{A),
corresponds to a morphfem a: P = F(l) >A. These various mor-

4-4 Characterization of monadic categories
217
St
^]1<^(P,B)^
9/
Pb
-^ B
f
Diagram 4.23
UJA) lUjB)
U{v)
Diagram 4.24
^ Q
phisms a yield a factorization ^ as in diagram 4.23 such that gos-^ = a.
In particular
fogosi = foa=pBOS^
since U{f){a) = 6, and thus f o g = p^. Therefore / is a regular epi-
morphism (see 2.1.6). So / reflects monomorphisms and regular epimor-
phisms, hence it reflects isomorphisms (see 2.1.6).
Let us now observe that U is an exact functor (see 2.3.5); it
preserves finite limits, since it has a left adjoint F (see 3.2.2, volume 1); we
must still prove that U preserves regular (or strong, see 2.1.4) epimor-
phisms (see 2.3.7). But this is exactly the assumption that P is
projective; indeed this means that given /: A >B, a regular epimorphism,
^{PJ): ^{P,A) >^{P,B) is surjective, or in other words [/(/) is
surjective.
It remains to prove condition B)(c) of 4.4.4. Consider thus a pair
u^v: A \ B in ^ such that (C/(tx), U{v)) admits a split coequalizer as
in diagram 4.24,
q o U{u) = qo U{v), qo s = 1q, U{u) or = lu(B) ? s o q =z U{v) o r.
We must prove that {u, v) has a coequahzer preserved by U. There is no
restriction in supposing the pair (tx, v) monomorphic, thus in supposing
that A is a relation on B see 2.5.1. Indeed consider diagram 4.25, which
is an image factorization in ^; see 2.1.4. This yields the relation pi o
i,P2 o i: T I n which has the same coequalizer as tx, ^, as long as

218
Monads
>BxB
Diagram 4.25
U(p) o r
/ U{pii) ^^
mi) I UJB)
U(p2i)
Diagram 4.26
U{Q)
one of those coequalizers exists, just because p is an epimorphism. Now
diagram 4.26 obviously remains a split coequalizer. So let us assume
that u^v: A ^B is a relation on B.
Since U preserves finite limits and reflects isomorphisms, it also reflects
finite limits (see 2.9.7, volume 1). So a relation ri,r2: R ^B in
IS
an equivalence relation in ^ if and only if C/(ri), U{r2) is an equivalence
relation in Set; see 2.5.4. Since U preserves finite limits, it preserves the
construction of the dual relation R^ (see 2.8.2) and since it is exact, it
preserves the composite of relations (see 2.8.3 and 2.3.6).
Let us come back to our relation u^v: A I B such that U{u), U{v)
has a spUt coequahzer. We consider the relation Ao A^ on B. This is
an equivalence relation, since U{A) o U{A^) is one in Set. First of all, if
beB, r{b) e U{A) and
r{b) = (wF),wF)) = [b,sq{b)).
In Set, the equivalence relation generated by U{A) is just the kernel pair
of q, thus contains U{A) o U{A^). If we prove that the kernel pair of q is
contained in U{A) o U{A^), we will have equality. But if q{b) = q{b') for
elements 6, b' G -B, then
{b,sq{b)) e U{A\ {sq{b\y) = {sq{b%b'),eU{A^)
from which F,6') G U{A)oU{A^). This proves that in Set, U{A)oU{A^)
is the equivalence relation generated by U{A), so that in ^, A o A^ is

4.4 Characterization of monadic categories 219
the smallest equivalence relation containing A.
Since ^ is exact, the equivalence relation AoA^ on B has a coequalizer
p, yielding an exact sequence
^1
AoA^ZZZZHXb ^—>P.
This exact sequence is preserved by [/, proving U(p) = q. It remains to
observe that p is also the coequalizer of {u^v). But given a morphism
/: B >C in ^ one has, by faithfulness of U and the relations q =
Coker {U{u),U{v)) = Coker (C/(ri),C/(r2)),
fou = fov^ U{f) o U{u) = U{f) o U{v)
^3h:Q >U{C) U{f) = hoq
^U{f)oU{r,) = U{f)oU{r2)
^ f ori = f or2. □
Let us conclude this section with an interesting link between monads,
comonads (the dual notion of a monad) and adjoint functors.
Proposition 4.4.6 Let T = (T, s, /i) he a monad on a category ^. If
the functor T: ^ >^ has a right adjoint, then the forgetful functor
U: ^^ >^ is comonadic for a comonad S = (S^p^S) on ^, where the
functor S is right adjoint to T.
Proof We shall apply 4.4.4. Observe that the reflection of isomorphisms
by U is asserted by 4.1.4 while the dual of condition B)(c) of 4.4.4 is
just a special case of 4.3.1. So it suffices to prove that U: ^^ >^ has
a right adjoint G: ^ >^^ such that [/ o G is right adjoint to T.
Let us write S\ ^ >^ for a right adjoint to T, with a: T o 5 =4> 1<^
and /3: 1<^ =4> S oT the two canonical natural transformations of the
adjunction T -\ S\ see 3.1.5, volume 1. Given an object D G ^, we must
construct its coreflection along U\ see 3.1.1, volume 1. The composite
TTS{D) ^^^^^ ) TS{p) —^^-^ D
corresponds by adjunction with a morphism a\ TS{D) >S{D); we
shall prove first that E(D), a) is a T-algebra. We refer heavily to 3.1.5,
volume 1, without mentioning it all the time; in particular the morphism
corresponding with a by adjunction is an o T{a)^ which proves that
ocD^T{a) = anOfisD' The first axiom for a T-algebra is croe'g^j-,^^ = 1s(d)''>
via the adjunction, this is proved by
OiD O fJ^S(D) O T{es(D)) = OLDO ItS(D) = OCD'

220 Monads
Next we must verify that aoT{a) = ao fisiD)] via the adjunction, this
is proved by
o^D o Ms(D) o TT{a) =aDO T{a) o fiTS(D) =o^d^ fJ'S(D) o fJ'TS(D)
Let us prove now that E(D), cr), together with the composite
S{D) ^^^^^ >TS{D) ^-P )I?,
is the coreflection of D along U. Given a T-algebra (C, ^) and a morphism
/: C >D^ we prove that the required factorization is given by the
composite
C—^^—^ST{C)^^^S{C)—^^^S{D).
First of all, this composite is a morphism of T-algebras from (C, ^) to
E(D),a); this means the equality
a o TS{f) o TSiO o T{l3c) = S{f) o S{0 o /3c o ^,
which follows by adjunction from the following relations:
ao o ixs{D) o TTSif) o TTSiO o TT{l3c)
= an o TS{f) o TSiO ° T{f3c) o fie
= /oacoT5(e)oT(/3c)o/xc
= / o ^ o aT{c) ° T{j3c) o Mc = / o ^ o Mc
= f oioT{i)^ f oioaT(c)oT{l3c)oT{i)
^foacoTS{OoT{Pc)oT{0
= ao o TS{f) o TSiO ° Til3c) o T(e).
On the other hand this morphism of T-algebras is a factorization of /
through ai) o Ss(d) since
c^D o es(D) o S{f) o 5@ of3c = aDO TS{f) o TS{0 o T{f3c) o sc
= fo^oaT{c) oT{f3c) o£c
= fo^osc
= /.
It remains to prove the uniqueness of that factorization. Given another
morphism h: (C,0 >{S{D),a) in ^ such that {ao o^s(d)) oh = f,

4.5 The adjoint lifting theorem
Q . ^
221
s/
U
V
> Si
R
Diagram 4.27
S{f) o 5@ o f3c = S{aD) o S{es(D)) o S{h) o 5@ o Pc
= Siao) o S{es(D)) o S{a) o ST{h) o Cc
= Siao) o ST{a) o S{£ts(d)) o ST{h) o f3c
= Siao) o S{fis(D)) o S{eTS(D)) o ST{h) o f3c
= S{aD) o ST{h) of3c = Sao o f3s(D) o h
= h.
Thus [/ has a right adjoint G: ^ >«''^ given by G{D) = E(D), a) on
the objects. But by 3.1.3, volume 1, if g: D >E is a morphism in ^,
G{g) is the unique morphism of ^^ such that the equality
{o^E o £s(E)) o G{g) = go{aDO es(D))
holds. Observing that
o^E o£s{E) o S{g) = aEOTS{g) o£s(d) =goaDOes(D)
we get G{g) = S{g). Therefore [7 o G = 5, which concludes the proof.
D
4.5 The adjoint lifting theorem
The present section is devoted to answering the following question:
Consider a commutative square of functors RoU = V o Q as
in diagram 4.27, where U and V are monadic. If R has a left
adjoint, does Q have a left adjoint?
The answer will be yes, provided s/ has (enough) coequalizers (see
exercise 4.8.5).
First of all, let us fix once for all the situation and the notation of
this section. We consider diagram 4.28, made of categories and functors
where T = (T, e, fjb) is a monad on ^; S = E, C» v) is a monad on ^; G, V

222
R
Diagram 4.28
are the corresponding forgetful functors from the categories ^^ and ^^
of algebras; and F, G are the free algebra functors. Choose R and Q such
that RoU = V o Q and L is left adjoint to i?, with canonical natural
transformations a: 1^ =^ Ro L and /3: Lo i? =4> l*^; see 3.1.5, volume 1.
The canonical natural transformations of the adjunction F -\ U are
s: leg ^ U o F and r: F oU =^ l^, where fi = U * r * F (see 4.2.2);
moreover T = FoU. In the same way the canonical natural
transformations of the adjunction G -iV are d^: 1^ =^ V o G and a: G oV =^ l^s,
where r/ = F * cr * G; moreover S = GoV.
By 4.1.4 we also know that if (C,^) is a T-algebra, then
^ = T(c,o: nC) = FC/(C,0 = (r(C),/ic)-
^(c^,0.
In particular, for every X G ^^ one has U{rx)ofjiu(x) = U{rx)oTU{Tx)^
by definition of a T-algebra (see 4.1.2). In the same way if F G ^^, one
has V"(c7y) O r7y(y) = V"(c7y) O 5V"(c7y).
To avoid too heavy a notation, in the rest of this section we shall use
notation like VaG to denote what should formally be written ly *cr* 1g;
for the same reason, we shall often omit parentheses, thus writing UTX
instead of UT{X) or VaGc instead of (ly * a * 1g)c-
Lemma 4.5.1 In the situation we have just described^ there exists a
natural transformation A: SR =4> RT which makes diagram 4.29
commutative.
Proof First of all define a natural transformation 6: GR => QF as the
composite
GR ^^ >GRT = GRUF = GVQF-^^^QF.
We define A: SR => RT as the composite
SR = VGR—y^-^VQF = RUF = RT.

R
4.5 The adjoint lifting theorem
CR
■^ SR^
TfR
RT ^
RH
Diagram 4.29
RTT<^
XT
-SSR
S\
■SRT
223
For the left-hand triangle, we have
Xo<:r^V6oCR = VaQF o VGRe o (^R
= VaQF o CRT o ite = VaQF o CVQF o ife
= Re.
And for the right-hand rectangle, we have
XorjR = V6oriR
= RHo ReT oV6orjR = Rfj,oXTo <^RT oV6orjR
= RHoXTo (^RT o VaQF o VGRs o rjR
^RnoXTo CRT o VaQF o SRe o rjR
= RHoXTo CRT o VaQF o rjRT o SSRe
^RfioXTo CRT o VaQF o rjVQF o SSRe
= Rfj.oXTo CRUF o VaQF o r?FQF o SSRe
^RfioXTo CVQF o FaQF o rjVQF o 55ite
= RnoXTo CVQF o FaQF o 5FaQF o SSRe
= RnoXTo SVaQF o SSRe
= RHoXTo SVaQF o 5FGite = RfjLoXTo SV6
^RfioXTo SX.
O
The reader will have observed that lemma 4.5.1 does not depend on
the existence of the left adjoint L to R. The same remark applies to the
next lemma.
Lemma 4.5.2 In the situation we have described, if{C, p) is a T-algebra,
the S-algebra Q{C, p) is given by {R{C), R{p) o Ac).
Proof Let us write Q{C,p) = {D,^). We have
D = V{Q), {C, p) = RU{C, p) = R{C).

224 Monads
On the other hand p = Ur^cp), C = U{C,p) and ^ = V[aQ(^c,p))'i thus
it remains to prove that VaQ = RUt o XU. Indeed
VaQ = VaQ o VGRUt o VGR£U = VaQ o VGVQt o VGReU
= l^Qr o FaQFC/ o VGReU = RUt o VaQFU o l^GiZeC/
= izc/r o XU n
Lemma 4.5.3 In the situation we have described, there exist natural
transformations u: FLS =4> FL and ip: G =^ QFL such that ip o aG —
Qoj o (fVG.
Proof Keeping the previous notation, we define a natural
transformation TT as the composite
LS ^^^ )LSRL ^^^ ) LRTL ^^^ > TL
and uj is then defined as the composite
FLS—^^!^-^FTL = FUFL '^^^ >FL.
On the other hand (p is the composite
G —^^-^ GRL —^^^-^ QFL.
To prove the stated equaUty, it suffices to prove that Vip o VaG =
VQu o VipVG, since V is faithful (see 4.1.4).
V(p o VaG = V6L o VGa o VaG = AL o 5a o 77
= AL o rjRL o SSa = RfiL o XTL o SXL o 55a
= RfiL o XTL o SRf3TL o 5ailTL o 5Ai: o SSa
= RuL o XTL o 5i2/3TL o 5ilLAL o SaSRL o 55a
= RuL o ATL o 5i?/3TL o SRLXL o 5i2i:5a o SaS
= i?^L o ATL o SRtt o 5a5
= RUtFL o ATL o 5i27r o SaS
= VQtFL o ATL o SRn o 5a5
= VQtFL o I^QFtt o ALFG o SaS
= VQu o XLVG o 5a5 = VQu o y^LFG o FGaFG
= VGcJ o FyjFG. D
The next two lemmas are not necessary for the proof of the adjoint
hfting theorem, but they will explsdn the construction used in that proof.

4.5 The adjoint lifting theorem 225
Lemma 4.5.4 In the situation described at the beginning of this section^
suppose further that Q has a left adjoint K. In these conditions^ uj =
KaG and (p = vG, where v: l^s ^ Q o K and 6. K o Q ^ l^^x are the
canonical natural transformations of the adjunction K -i Q.
Proof Prom RoJJ = V o Q^ we deduce the corresponding relation
FoL = KoG between the left adjoints; there is no restriction in assuming
- via the axiom of choice - that the left adjoint K has been chosen so
that the equality FoL = KoG holds. It is immediate from 3.2.1,
volume 1, that the canonical natural transformations of the composite
adjunction K oG -\V oQ are
1^ ^ >VG ^^^ yVQKG, KGVQ ^^^ yKQ ^—>1^t
while those of the adjunction F o L -\ RoJJ are
1^ ^—>RL ^^ yRUFL, FLRU ^^^ > FU ^ >V.
In particular ReL o a = VuG o ^ and r o F/3C/ = 6 o KaQ. Let us first
observe that VuG = XLo5a, i.e. VuG = V6o VGa. Via the adjunction
G H F, we know that iyG = 6oGaiiand only if VuGo^ = VSoVGaoC,
which is the case since by lemma 4.5.1 and previous relations
VuG o C = ReL oa = XLo (^RL oa = XLoSaod;.
By 3.1.3, volume 1, KaG is the unique natural transformation such
that QKaG o i/GVG = uG o aG. So we must prove that Qu o i/GVG =
uG o aG, which is equivalent to VQu o VuGVG = VuG o VaG, since
V is faithful. As just observed, this reduces to VQu o XLVG o SaVG =
XLo Sao VaG. And indeed, by lemma 4.5.1,
VQuj o XLVG o SaVG
= RUu o XLS o SaS = RUrFL o RUFn o XLS o SaS
= RfiL o RUFtt o XLS o SaS = RfiL o XTL o SRtt o SaS
= RfiL o XTL o SRf3TL o SRLXL o SRLSa o SaS
= RfiL o XTL o SRf3TL o SRLXL o SaVGRL o SSa
= RfiL o XTL o SRf3TL o SaRTL o SXL o SSa
= RfiL o XTL o SXL o SSa
= XLo rjRL o SSa = XL o Sa o rj
= XLo Sao VaG.

226 Monads
Again since V is faithful, it suffices to prove Vcp = VuG to get ip = vG.
And indeed
V^ = V6L o VGa = XLoSa = VuG, D
Lemma 4.5.5 In the situation described at the beginning of this section,
suppose further that Q has a left adjoint K. Under these conditions,
given a S-algebra (D, ^), K{D, ^) is given in ^^ by a coequahzer diagram
of the form
FLS{D) lFL{D) >K{D,(,)-
Proof In ^^, the coequalizer diagram of lemma 4.3.3 can be written,
with our notation, as
(tGd fj
GS{D) ]G{D) ^^-^^(AO-
Now K preserves this coequalizer, since it has a right adjoint Q.
Moreover K o G = F o L since RoU — V o Q and KgGd = ^d by lemma
4.5.4. Therefore we get the stated coequalizer in ^^. D
We are ready to prove the adjoint lifting theorem.
Theorem 4.5.6 Let RoU = VoQbea commutative diagram of
functors, where U and V are monadic, as in diagram 4.27. If j/ has
coequahzers, then Q has a left adjoint as soon as R has a left adjoint.
Proof We use the notation defined at the beginning of this section
and in lemmas 4.5.1 and 4.5.3. Given a S-algebra (-0,0^ we define a
T-algebra X by the following coequalizer in ^^:
FLS{D)^^^FL{D) ^ >X;
compare with 4.5.5. Moreover by lemma 4.5.3, the left-hand part of
diagram 4.30 is commutative while the top row is a coequalizer
diagram by 4.3.3. This implies the existence of a unique factorization x
such that Q{x) o <^d = X ^ ^- We shall prove that X, together with
X' (i^,0 ^QX, is the reflection of (D,^) along Q.
Choose a T-algebra (C, p) and a morphism y: (D, ^) >Q{C, p) of T-
algebras. By 4.5.2, we know already that Q{C,p) = {R{C),R{p) o Ac),
thus in particular we get a morphism V{y): D >R{C) such that the
relation R{p) o Ac o SV{y) = V{y) o ^ holds. Let us then consider the
composite in ^^,
FLiD) ^^^^y\FLR{C) ^^^^) >FiC) ^-^{C,p),

4-5 The adjoint lifting theorem 227
GSD "" ] GD^^^^iD,^
^SD
^D
Qojd
—>
—>
Diagram 4.30
QFLSDZZ^^QFLD—^—> QX
QFLi ^^
and prove it coequalizes ujd and FL^. Since U is faithful, it suffices to
prove the equality
po UFf3c o UFLVy o Uud =po UFf3c o UFLVy o UFL^,
Indeed
p o UFCc o UFLVy o Uujd
= po UFi3c o UujRc o UFLSVy
= po UFi3c o UtFLRc o UFnRc o UFLSVy
= po TCc o fiLRc o UFnRc o UFLSVy
= poficoTTCc o UFnRc o UFLSVy
= poficoTTf3c oTf3TLRc o UFLXLRc o UFLSaRc o UFLSVy
= pofico Ti3Tc oTLRTPc o UFLXLRc o UFLSaRc o UFLSVy
= poficoTf3Tc oTLXc oTLSRf3c o UFLSaRc o UFLSVy
= pofico TpTc o TLXc o UFLSVy
= poTpoTf3Tc oTLXc o UFLSVy
= po Tf3c o TLRpo TLXc o UFLSVy
= po Tf3c o TLVy o TLi
= po UFPc o UFLVy o UFL^.
By this coequalizing property and the definition of X, we get a unique
morphism z: X >(C,p) in ^'^ such that zox = po Ff3c o FLVy.
We must prove that z is the required factorization, i.e. the unique
morphism of T-algebras such that Qz ox = y- Since ^: CD >{D,^)
is a coequalizer in ^^, this is equivalent to Qzoxo^ = yo^. And since
V is faithful this is further equivalent to VQz o Vx^^ = Vy o^. And in
fact

228 Monads
VQzo Fx o ^
= VQz o VQx o VifD = VQp o VQFPc o VQFLVy o Vipo
^Rpo VQFf3c o VipRc o VGVy
^Rpo VQFl3c o VSLRc o VGaRc o VGVy
^Rpo V6c o VGRf3c o VGaRc o VGVy
^Rpo V6c o VGVy = RpoXco VGVy
= Vyo^.
It remains to check the uniqueness of the factorization. Given another
morphism w: X > (C, p) of T-algebras such that Qw o ^ = y, we must
prove that z = w oi^ equivalently, z ox = w ox since x is a coequalizer.
Since ^ has a section e^^ this is also equivalent to proving zoxoFL^ =
woxoFL^ or, by faithfulness of C/, to UzoUyoUFL^ = UwoJJxoUFL^.
Indeed, writing X = (UX^tx)-, w: (UX^tx) >(C',p) is a morphism
of T-algebras and one has
UzoUxo UFLi
= po UFf3c o UFLVy o UFL^
= po UFf3c o UFLVQw o UFLVx o UFL^
= po UFf3c o UFLRUw o UFLVx o UFL^
= po UFUw o UFf3UX o UFLVx o UFL^
= po UFUw o UFPUX o TLVQx o TLV^d
= po UFUw o TUx o Tf3TLD o TLVipo
= po UFUw o TUx oT^TLd o UFLV6Ld o UFLVGao
= po UFUw o TUx o TPTLd o UFLXLd o UFLVGao
= poTUw oTUx o UFttd = Uwo UrX oTUx o UFttd
= Uw oUxo UtFLd o UFttd = Uw oUxo Uujd
= UwoUxo UFLi. D
Corollary 4.5.7 Let U = V oQ be a commutative triangle of functors,
where U and V are monadic, as in diagram 4.31. If s^ has coequalizers,
Q is monadic as well.
Proof In 4.5.6 put R = 1<^, which yields the existence of a left adjoint
to Q. It is now easy to get the conclusion by applying 4.4.4. First, Q
reflects isomorphisms, because V preserves them and U. reflects them.
Now if a pair tx, v: X >y is such that Q{u),Q{v) has a split
coequalizer p: Q{Y) >P in ^, then {U{u),U{v)) = {VQ{u),VQ{v)) has a

4-5 The adjoint lifting theorem
Q
229
Diagram 4.31
-> S
SoS ^*^ ->ToT
M
^ T
Diagram 4.32
split coequalizer in <^ and thus {u,v) has a coequalizer z: Y >Z
preserved by U. If to: P >Q{Z) is the unique factorization such that
w op = Q{z), then both U{z) — VQ{z) and V{p) are coequalizers of
{U{u), U{v)) in <<^; therefore U{w) is an isomorphism and thus w is an
isomorphism. Thus the coequaUzer z of (tx, v) is preserved by Q. D
The previous result, via lemma 4.5.1, is in close relation with the
notion of a "morphism of monads".
Definition 4.5.8 Let T = (T^e^fi) and S = E, C,^) be two monads
on a same category ^. By a morphism of monads S ^-T, we mean a
natural transformation A: 5 => T such that Xo(^ = e^ fio{XoX) = Xorj;
see diagram 4.32.
Proposition 4.5.9 Let T = (r,£:,/i) and S = (SX^rj) be two monads
on a same category ^. There is a bijection between:
A) the morphisms of monads X: S ^-T;
B) the functors Q: ^^ >^^ such that V oQ = U, where U and V
are the corresponding forgetful functors (see diagram 4.33).
Proof By lemma 4.5.1 (putting R = 1*^), every functor Q such that
VQ = U induces a natural transformation X: S ^ T with the required
properties. Indeed (A * X)c = Xt{c) o S{Xc) = T{Xc) o Xs{c)^ by 1.3.4,
volume 1.
Conversely given a natural transformation X: S => T as in 4.5.8, we
define a functor Q: ^ >^ by (see lemma 4.5.2)

230 Monads
Q
Diagram 4.33
• Q(C, p) = (C, p o Ac) for a T-algebra (C, p),
• Q(/) = / for a morphism /: (C,p) >{D,^) of T-algebras.
(C, p o Ac) is an S-algebra because
(p o Ac) o Cc = P o ^c
(po Ac) o r/c = Po Mc o (A * A)c = po fic o Xt(c) o 'S'Ac
= p o r(p) o Xt{c) o 'S'(Ac) = p o Ac o 5(p) o 5(Ac)
= (poAc)o5(poAc),
and Q{f) is a morphism of S-algebras because
/o(poAc)=^or(/)oAc = (^oA,,)o5(/).
On the other hand, the functoriality of Q is obvious as well as the relation
VoQ = U.
It remains to prove that the two constructions are mutually inverse.
Let us start with A and construct Q. With the notation of 4.5.1 we have
F{C) = {T{C),fic) and thus aQFc = fJ^c ^ Xtc- Therefore the natural
transformation X: S =^ T constructed from Q as in 4.5.1 is such that
A^ = VaQFc o VGec
= fjic ^ Xtc o Sec = Mc o Tec o Xc
= Ac.
Conversely start from Q such that VQ = U and construct A as in
4.5.1. Construct now Q from A as at the beginning of the proof, thus
Q(C,p) = (C,p o Ac). By lemma 4.5.2, Q(C,p) = Q(C,p). The case of
morphisms is obvious. D
Corollary 4.5.10 Let T and S be two monads on the category of sets.
There are bijections between:
A) the morphisms of monads S >T;
B) the functors Q: Set"^ >Set^ such that VoQ = U;

4-6 Monads with rank 231
/ U{u) '^ ^ ^
U{X) lUjY) ^-^ Q
U{v)
Diagram 4.34
C) the monadic functors Q: Set^ ^-Set^ such that VoQ = U;
where U: Set^ >Set and V: Set^ ^-Set are the corresponding
forgetful functors.
Proof The result follows from 4.3.5, 4.5.9 and 4.5.6. D
4.6 Monads with rank
Let us first introduce some classical terminology.
Definition 4.6.1 A monad T = (T^e^fi) on a category ^ has rank a,
for some regular cardinal a, when the functor T: ^ >^ preserves a-
filtered colimits. When a = i<o? ^hus when T preserves filtered colimits,
one also says that T has finite rank.
Proposition 4.6.2 There is a coincidence between;
A) the categories ModT of models of an algebraic theory T, in the sense
of 3.3.1;
B) the categories of T-aigebras, where T is a monad with finite rank on
the category on sets.
Proof Let T be an algebraic theory in the sense of 3.3.1. By 3.9.1,
the forgetful functor U: Mod^? >Set has a left adjoint and reflects
isomorphisms. In order to apply 4.4.4, let us also consider a pair of
morphisms u, v: X ^Y in Modr such that {U{u), U{v)) admits a split
coequalizer as in diagram 4.34:
qoU{u) = qoU{v), U{u) o r = lu^y), qos = lQ, U{v) or = soq.
Raising this diagram to powers n, m G N yields other coequalizer
diagrams (see 2.10.2, volume 1) so that every arrow a G T{T'^,T'^) yields
a corresponding factorization Q{a): Q^ >Q^ as in diagram 4.35,
making Q a T-model and q: Y >Q a morphism of T-models. By
construction q o u = q o v; on the other hand, if w: Y >Z in Modr is
such that w o u = w o v^ we get in the category of sets a unique map
t: Q >U{Z) such that toq = w. To prove that t is a T-homomorphism,

232
Monads
x(r^) ^yfr^v
Vn
X{a)
Un
Y{a)
X{T^) \Y{T^) ^^ ) Q^
Diagram 4.35
we must check that Z{a) o T = t^ o Q{a) for every a G T{T'^,T'^).
Since g^ is a surjection, this follows from
Z(a)orog^ = Z(a)o'w;^ = w'^oYia) = t^og^oy(a) = t^oQ(a)og^.
Thus g: y >Q is the coequalizer of (u^v) in Modr and by 4.4.4,
U: Mod^T ^-Set is monadic. If F is left adjoint to [/, the
corresponding monad admits UF as functor part and UF preserves filtered colimits
(see 3.9.1).
Conversely, let us suppose that T is a monad with finite rank over Set.
Using 4.1.4, 4.3.2 and 4.3.5, we can conclude the proof by 3.9.1. D
Corollary 4.6.3 Every algebraic functor, in the sense of 3.7.4, is
monadic and the corresponding monad has finite rank.
Proof By 3.9.2 and 4.5.7, every algebraic functor is monadic. By
3.7.5, the corresponding monad has finite rank, since an algebraic functor
preserves filtered colimits while its left adjoint preserves all colimits (see
3.2.2, volume 1). D
Counterexample 4.6.4
The category of abelian groups is monadic over the category of sets (see
4.6.2). On the other hand the category of torsion free abelian groups
is a localization of the category of abelian groups (see section 1.12 and
theorem 1.13.5), thus is monadic over the category of abelian groups
(see 4.2.4). But the composite functor
U
-^Set
Torsion free abelian groups C Abelian groups
is not monadic. Indeed the left adjoint to U is the free abelian group
functor ... and every free abelian group is certainly torsion free! Thus
the compoi^te functor again admits the free abelian group functor as

4.6 Monads with rank 233
a left adjoint and the corresponding monad therefore admits the
category of abelian groups as category of algebras. Therefore the composite
functor is not monadic (see 4.4.1 and 4.2.2).
Replacing 'Rq by an arbitrary regular cardinal a, one can consider
the a-algebraic theories T defined as categories with objects r^(n < a),
where T^ is the n-th power of T^; a T-model is then a functor T ^-Set
which preserves a-products. Proposition 4.6.3 generalizes easily to give
a coincidence between the a-algebraic categories and the categories of
T-algebras for a monad T of rank a on the category of sets.
But not every monad on the category of sets has a rank. Here is a
classical and easy counterexample
Proposition 4.6.5 Let ^ be the category of \/-lattices, i.e. the objects
of ^ are the complete lattices and the morphisms are the mappings
preserving arbitrary suprema. The forgetful functor is monadic over the
category of sets but the corresponding monad does not have a rank.
Proof Consider the covariant power set functor T: Set > Set mapping
a set X to its power set V{X) and a mapping /: X >Y to the direct
image mapping
V{f):V{X) >V{Y), Z^fiZ).
One also defines e: Igg^ =^ T by
ex-.X >V{X), x^{x}
andfi-.ToT^Thy
lix: V{V{X)) >V{X), {Zi)iei ^ Ui^/i.
It is easy to check that T = (T, £:, /i) is a monad.
Let us now define a comparison functor J: ^ ^-Set just by:
• J{C) = (C, V), where V- T{C) >C maps a subset Z C C to its
supremum;
• J(/) = f a f: C >D is a morphism of ^.
The two relations V{^} = x ioi x G C and
\/{\/Zi\ZiCC, iG/}=V(U,/0
indicate that J{C) is a T-algebra. On the other hand J(/) is a morphism
of T-algebras just because / preserves suprema.

234 Monads
More generally, observe that a mapping g: J(C) >J{D) is a mor-
phism of T-algebras precisely when g preserves suprema; thus J is full
and faithful.
Finally given a T-algebra (X,^), define x < y when ^[{x,y}) = y.
Prom ^({x,x}) = ^{{x}) = {^o ex){x) = x, we get x = x.Ii x <y and
2/ < X, we have ^({x,t/}) = y and C({2/)^}) = ^^ thus x = y. Next if
X < 2/ and y < z, the relations C({^) v}) = V ^^^ ^{{v^ ^}) = ^ imply
ax,z} = ^{^{x,x},^{y,z}} ^ {^oT{0){{x,x},{y,z}}
^^°t^x{{x,x},{y,z}} =^{x,y,z}
= (eo/ix){{x,y},{2/,4} = (|o7^@){Ry},{2/,4}
= eU{x,y},e{y,4} = e{y,4
= 2;;
thus x < z.So (X,^) is already a poset. Now ^: T'(X) >X is just the
supremum operation for this poset structure. Indeed, given Z C X and
xeZ,
^{x,aZ)} = ^{a^UiZ)}-{^oT{0){{x},Z}
^{^of,x){{x},Z}^^{{x}uZ)
=az);
thus X < ^{Z). Next, if y G X is such that x < y for every x G Z,
mzU} = mz).ay}} = {^oT{o){zdy}}
= ^{y\xez} = ay}
= y\
hence ^(Z) < y. We have thus checked that ^(Z) is just the supremum of
Z, which indicates that (X,^) is just J(X, <). This concludes the proof
that ^ is equivalent to Set^.
Now let us observe that the functor T: Set ^-Set does not preserve
a-filtered colimits, whatever the regular cardinal a. Just choose a set X
whose cardinal is at least a and write X as the union of all its subsets Y
with cardinality strictly less than a; this is an a-filtered union, because
a is regular, thus an a-filtered colimit in the category of sets (see 4.13.3).

4.6 Monads with rank 235
Applying T to that colimit does not yield a a-filtered colimit. Indeed for
every inclusion i: Y >X in the original diagram and every Z G T{Y)^
Ti{Z) has cardinality strictly less than a. In particular X G T{X) cannot
be presented as the equivalence class in the colimit of an element Ti{Z),
which proves that the second diagram, which is still a-filtered, does not
admit T{Z) as a colimit (see again 4.13.3 and the construction of the
a-filtered colimit colimyr(y) in 6.4, volume 1). D
Clearly it would have been faster to apply 4.4.4 to prove the monadic-
ity of the category of \/-lattices; we found it more instructive to give an
example of explicit computations with a monad.
Comparing 4.6.5 and 4.6.2, we can say that the theory of \/-lattices
has operations of arbitrarily large arity: for every cardinal /3 and every
/3-family of elements, there must exist a "composite" of that family and
the axioms force this composite to be exactly the supremum of the family
for a poset structure. This should be compared with examples 3.3.5.C,
d, e, f.
With the previous remark in mind, one might now wonder if a theory
which can be described via operations of arity /3, for every cardinal /3,
and axioms expressed by equalities, is necessarily induced by a monad
on Set. The answer is no (see exercise 4.8.9).
Let us conclude this section with another interesting example of a
monadic category over Set (in fact, a monad without rank).
Proposition 4.6.6 The category of compact HausdorfF spaces is
monadic over sets.
Proof Write Comp for the category of compact Hausdorff spaces and
U: Comp ^-Set for the underlying set functor. We apply 4.4.4 to prove
that U is monadic.
In 3.3.9.C, volume 1, we proved that the inclusion Comp C Top of Comp
in the category of topological spaces has a left adjoint (the Stone-Cech
compactification). In 3.1.6.J, volume 1, we proved that the underlying set
functor Top >Set has a left adjoint (the discrete topology functor).
By 3.2.1, volume 1, our functor U: Comp >Set thus has a left adjoint.
A continuous bijection between compact Hausdorff spaces is
automatically a homeomorphism; therefore U reflects isomorphisms.
Finally consider two continuous mappings u^v: XZZ^y withX, Y
compact Hausdorff spaces and suppose they have a split coequalizer in
the category of sets, as in diagram 4.36:
qos=lQ, txor=ly, qou== qov, vor = so q.

236 Monads
X I Y ^-^ Q
V
Diagram 4.36
Let us provide Q with the quotient topology, induced by the topology
of y. If we prove that Q is Hausdorff, Q will be compact Hausdorff as
continuous image of the compact Hausdorff space Y and q will be the
coequalizer of (u^v) in Comp by 2.4.6.e, volume 1.
This is equivalent to proving that the kernel pair of g is a closed
equivalence relation inY xY; see 2.4.6.f, volume 1.
Write R for the relation on Y defined by
R= Uu{x),v{x))\x e x\;
it is the image of X under the continuous mapping ( j : X >Y x y,
thus is compact, since X is compact, and therefore closed, since Y is
Hausdorff.
Now choose y gY. One has immediately
{y^sq{y)) = {ur{y),vr{y)) G R.
Therefore if y,y' ^Y with q{y) = q{y')
{y,sq{y)) G R and {sq{y'),y') G i?^ ^ {y',y) G i?^ o i?
where R^ is the inverse relation of R (see 2.8.2) and Ro R^ is the
composite of the relations R^R^, as defined in 2.8.3. Conversely given a
pair {y\y) G i?^ o i?, there exists y^^ G Y such that {y,y^^) G R and
{y" '^y') ^ R-> thus q{y) = q{y'') = q{y') since qo u = qov. This proves
that R^ o Ris the kernel pair of q.
By 2.8.3, the composite relation R^oRis given in Set by diagram 4.37
where the square is a puUback and the triangle is an image
factorization. Since U: Comp >Set preserves limits (see 3.2.2, volume 1), the
pullback can equivalently be computed in Comp, but also the product
Y xY. Since a continuous image of a compact space is compact, RP oR
is compact in the HausdorflF space Y xY^ thus is closed. D

J^.l A glance at descent theory
237
a
R
13
P2
-> R^
Pi
-> Y
Diagram 4.37
■^YxY
4.7 A glance at descent theory
->^ is a monadic functor with left adjoint functor F: ^-
->j/
IfC/:j/-
(see 4.1.4), descent theory along U is intended to characterize the objects
of the form F{B) as objects of j^ provided with a convenient structure,
and analogously for morphisms. We recall here some basic facts about
the classical case of modules over a ring and show how the "descent
data" can be elegantly described using the comonad on j^ associated
with the adjunction F -\U. We assume some familiarity with classical
module theory.
Throughout this section, we fix a homomorphism /: R >S of
commutative rings with unit. We consider the corresponding algebraic
functor (see 3.9.3)
/*: Mod^-
->Modi?, M \-^ M
where scalar multiplication on M G Modi? is given by r • m = /(r) • m
for r G i?, m G M. It is well known that this functor admits as a left
adjoint
/i: Modi? >Mod5, N ^ N®rS
where, in the tensor product, S is provided with the i?-module structure
given by 5 • r = 5 • /(r) for 5 G 5, r G R.
Let us also consider the ring S®rS^ where the multiplication is
induced by
{Si ® S2){s\ ® s'2) = Sis[ ® S2S2'
This yields the following diagram in the category of commutative rings
with unit:
R L >g IS®rS i^—>S,
rJ2

238 Monads
where the various new ring homomorphisms are determined by
rii{s) = s®l, rJ{s) = 1<S> S, fjL{s (S> s^) = ss\
Just by definition of 5(8)^5, one has rjio f = rj20 f.
Consider now an 5-module M. So {rii)\{M) is the E(8)i?5)-module
M(S)s{S(S)rS) ^ {M®sS)(SirS ^ M<^rS
with scalar multiplication induced by
(m (g) {s[ <S> 4)) • (^1 <S> S2) = m(S> {s[si (g) 52^2)
together with the requirement that
m® E (g) 5') = m (g) (E (g) 1)A (g) 5')) = m(g) G71E) • A (g) 5'))
= 7715® (l(gM0-
So, viewing G7i)!(M) as M®rS, scalar multiplication is determined by
{m (g) 5)Ei (g) 52) = 7n5i (g) 552.
We shall often write M(SirS instead of {rji)\{M).
In an analogous way we can consider, for an 5-module M, the {S®rS)-
module G72I (M) which is just
M(S)s{S(S)rS) ^ {M(SisS)(S)rS ^ M®rS
with scalar multiplication induced by
{m (g) {s[ (g) 53)) E1 (g) 52) = 7n (g) Ei5i (g) 5352)
together with the requirement that
7n (g) E (g) 5') = m (g) (A (g) 5')E (g) 1)) = 7n (g) G72E') • E (g) 1))
= ms^ (S) E® 1).
So, viewing G72I (M) as M(S>rS, scalar multiplication is determined by
{m (g) 5)E1 (g) 52) = 7n52 <S> 55i.
As the tensor product over R is symmetric (up to isomorphism), it is
more sensible to write G72I. (M) = S®rM together with the
multiplication
E (g) m){si (g) 52) = 55i (g) 7n52.
We shall often write S(S^rM instead of {rj2)\{M).
Now if iV, iV' are two ii-modules, let us write
CTNN'' N(S^rN^ >N^(SirN

4.7 A glance at descent theory 239
for the symmetry isomorphism determined by (Jnn' (^ ® ^0 = n' ® n; it
is an i?-linear mapping. In particular given an 5-module M, we get an
i?-linear isomorphism
(JMS- {vi)\{M) >G72I (M).
Let us insist on the fact that aMS is only i?-linear, even if {r)i)\{M) and
G72I (Af) are 5(8)i?5-modules.
To end the fixing of notation, let us still consider, for an 5-module M,
the following i?-linear mappings:
{rii)\{M) = M(S)rS-^^M, m®s^ms,
{rJ)\{M) = S®rM-^^^M, s®m^ms.
With our notation settled, we come back to the descent problem. We
want to investigate when an 5-module M has the form f\{N)^ for some
i?-module N. Since 771 o / = 772 o /, we get an isomorphism {r)i)\f\{N) =
{V2)\f\{N) for every i?-module N. This isomorphism of E(8)i?5)-modules
{m)\MN) ^ {N®rS)®s{S®rS)^^{N®rS)®s{S®rS) ^ {ri2)\MN)
is just determined by
(fNiin ® s)® {si ® S2)) = {n® s)® E2 ® Si).
Indeed, (pN is correctly defined since, given an element 5' E 5,
(fN {{'n ® 5M' ® {si ® S2)) = (fN {{'n ® ss^) ® {si ® S2))
= {n® 55') ® {s2 ® Si)
= {{n®s)s^) ®{s2®si)
= {{n ®s)®{l® s^){s2 ® si))
= {n® s) ® {s2 ® s^si)
= ^n{{ji ®s)® {s'si ® 82))
= (fN{{n ®s)® E' ® l){si ® 52)),
and obviously (f^ is an isomorphism with inverse defined by the same
formula. Via the previous isomorphisms
{N®rS)®s{S®rS) ^ {N®rS)®rS,
{n® s) ® {si ® S2) ^-^ n® ssi ® 52,
{N®rS)®s{S®rS) ^ S®r{N®rS),
{n®s)® {si ® S2) ^-^ Si®n® 552,

240
{N®rS)®rS-
Monads
<t>N
^S®r{N®rS)
crN<s>RS,s
S0r{N®rS)
^N^rS
^NiS>rS
{N(^rS)®rS®rS
> N<S>rS
1 ® crsSy
0iV ® 1
{N(^rS)(^rS(^rS
0iV ® 1
S<^r{N(^rS)(^rS-
1 ® crN<s>RS,s
Diagram 4.38
S®r{N®rS)<^rS
l®(t>N
'S®rS®r{N®rS)
used to calculate {rii)\{N(S)RS) and {rj2)\{N(SiRS), our isomorphism (pN
becomes
(m)!/!(iV) = (Ar®fi5)Ofl5—^^^^5®fl(iV®fl5) ^ (r?2)!/!(iV)
and is determined by
0iv(^(8) 5i (g) 52) = 5i (g)n® 52.
It is now an obvious matter to check the commutativity of both parts
of diagram 4.38. In the first one, an element n ® Si ® 82 is mapped to
n(gMi52 while in the second one an element n® 5i (gM2 053 is mapped
to 5i (g) 52 (g) n(g) 53. Moreover given an i?-linear mapping g: N >N
and considering the corresponding isomorphism </>^ obtained from iV,
we get the commutative diagram 4.39, where an element n(g) 5i (g) 52 is
mapped to 5i (g) /(n) (g) 52.
Definition 4.7.1 Let f: R >S be a homomorphism of commutative
rings with units; we use the previous notation. By a descent datum on
an S-module M we mean an {S(S>RS)'hnear isomorphism
</>: M(SirS >S<S>rM

J^.l A glance at descent theory
(f <8> 1) <8> 1
241
4>N
S®{N®rS)-
<hj
N
1®(/®1)
Diagram 4.39
■>5 ® (iVOfl5)
M®rS-
<t>
->S®rM
<^MS
S<S)rM
Am
» M
1 iSxJss,
M<S>rS<S>rS
0(8I
4^
S<S>rM(^rS
^®cfms
Diagram 4.40
S®rM®rS
1(8H
4^
S®rS®rM
which makes the two parts of diagram 4.40 commutative. If(j) is a descent
datum on the S-module M and 0 is a descent datum on the S-module
M, by a morphism g: (M, (/>) > (M, (/>) of descent data we mean an
S'linear mapping g: M >M making diagram 4.41 commutative.
The previous considerations immediately imply the following result.
Proposition 4.7.2 Let f: R >S be a homomorphism of commutative
rings with units. Write Des(/) for the category of descent data described
in 4,7.1, There exists a "comparison" functor
k: MocIh >Des{f)

242 Monads
0
0
S®rM-j^S0rM
Diagram 4.41
mapping a R-module N to {N(S)rS, (J)n) Sind correspondingly an R-linear
mapping g: N >N to g® 1^. D
Definition 4.7.3 A homomorphism f: R >S of commutative rings
with unit is called an effective descent morphism when the comparison
functor
ModR ^Des(/)
of 4.7.2 is an equivalence of categories.
In the case of an effective descent morphism /: R >S, the answer
to our original question is thus: an 5-module M has the form N(S)rS
for some i?-module N if and only if M can be provided with a descent
datum.
Here is now the key result, connecting descent theory with the theory
of (co)monads.
Theorem 4.7.4 Let f: R >S be a homomorphism of commutative
rings with units. Consider the corresponding adjunction
Modfi^ >Mod5, /iH/*,
/!
where f*{M) = M and fi{N) = Nf^sR- Write T = {T^e^fi) for the
comonad on Mods induced by this adjunction (see 4.2.1). The category
Des(/) of descent data for f is equivalent to the category Mod5 ofcoalge-
bras for the comonad T; moreover^ via this equivalence^ the comparison
functor of proposition 4.7,3 is just the comparison functor refered to in
4,2,1,
Proof Using the construction of 4.2.1, the comonad T can be described
as follows. The functor T is given by
T: Mods >Mods, M y-^ M(S)rS,

4-7 A glance at descent theory 243
where the 5-module structure on M<S>rS is just
G71(8MM' = m<S> ss\
The two natural transformations £:, /i are given by
eiT^id, Em'-M®rS >M, m®s\-^ms,
fi:T ^ToT, fiM'M<S>rS >M<S)rS<S)rS, m<S)S \-^ m<S)l<S) s.
The axioms for a comonad are obviously satisfied by £:, /i.
A T-coalgebra is a pair (M,^) where M is an 5-module and the S-
linear mapping ^: M >M(S>rS satisfies the conditions dual to those
of 4.1.2. The 5-linearity of ^ means
\/meM \/seS i{ms) = i{m)'{l®s).
Now writing ^(m) = XlILi^* ® ^»' ^^^ ^^^ axioms for a T-coalgebra
become
MmeM ^._ ^(mi) ® ^i = ^._ m^ (g) 1 (g) 5^,
En
mi5i.
i=l
Starting with such a T-coalgebra (M, ^), let us construct a descent
datum 0: M®rS >S®rM. With the previous notation, (j) is the
composite
M®rS ^®^ )M®rS®rS^®^^^^M®rS®rS ...
... Pm®^,M®rS ^^^ )S®rM.
The action of this composite on an element m® 5 is thus given by
En w--vn
rrii ® Si ® s \-^ y rrii ® s ® Si...
En y—^n
rriiS® Si\-^ y. Si®miS.
Observe that (f) is E(8)H5)-linear since, given a,/3 in 5,
(^ ® l){am ® /3s) = ^{am) ® f3s = ^(m) '{l®a)® f3s
= (Xl''_ ^^ ® ^v • A ® ^) ® /^^
En
mi (g) 5ia (g) /35,
from which one deduces immediately that
<f>{am ® Ps) = ^2 - ^'^^ ^ mi/35 = {y^ - ^^ ® ^*^) ' i^® P)
= (ff{m®s) • (a0^).

244 Monads
To prove that (/> is an isomorphism, just observe that it admits for
inverse the following composite xp:
S®rM ^ ^ ^ ) S®rM®rS ^^^ ® \ M®rS®rS P^®^) M®rS.
Indeed, given 5 G 5 and m G M,
A ® 0(t>{m (8) 5) = A (g) 0 (Xlili^^ ® ^^^) ^ XlJ-i^^ ® ^(^i^)
= 5]]J^^5i(8)(^(mi)-(l(8M))
= ^._ Ei (g) TTli (g) 1) • A (g) 1 (g) 5)
En
5i (g) TTli (g) 5
i=l
from which it follows immediately that.
En
mi5i (g) 5 = m(g) 5.
The relation @ o '^)E (g) m) = 5 (g) m is proved in a completely analogous
way.
The two conditions for a descent datum (see 4.7.1) are proved by the
same type of computation. The first condition is just obvious since
(Am o(j))(rn® s) = Am (y^._ ^i ^i^iS) = /J.__ SiUiiS = ms
(Am o (^ms){'^ ® s) = AmE ®m) = sm.
And the pentagonal condition is proved in the following way:
A ® (j)){(l) ® l)(m ®s®t) = {l® (l>){(l>{m ®s)®t)
= A (g) 0) (^._ Si ® rriiS ® tj
= y^ Si®(l>{miS®t).
To compute (f){miS (g) t), we observe that
V". Si®{^®l){miS®t) = y^ ^ Si®i{mis)®t
= Yl - ^i ® ^(^*) • A <^ 5) ^ ^
= ^.__ Ei(g)^(mi)(g)l)(l(g)l(gM(g)t)
= ^._ {si®mi®l®l){l®l®s®t)
En

4-7 A glance at descent theory 245
from which it follows that
A <S> (/>){(!> ®l){m®s<S>t) = y^ Si®s® rriit.
On the other hand
A (g) aMs){(t> ® 1)A ® crss){rn (g) 5 (g) t)
= A (g) <TMs){(t> ® 1)(^ ®t®s)
= A (g) gms) {(t>{m (g) t) (g) 5)
= A (g) (Jms) {y^ - ^'' ® ^*^ ® ^)
= ^._ 5i (g) 5 (g) TTlit.
We have already proved that starting with a T-coalgebra (M,^), the
morphism 0 we have defined is a descent datum on M. Given another
T-coalgebra (M,^), consider the corresponding descent datum cj). Given
a morphism of T-coalgebras g: (M, ^) > (M, ^), let us observe that
g: {M^(j)) >{M^(j)) is also a morphism of descent data, which will
imply immediately that we have constructed a functor
0: Mod^ >Des(/).
Given 5 G 5 and m € M, we must prove that
A (g) g)(i){m ®s)= '4>{g ® l)(m ® 5),
while our assumptions are
g{ms) = 5f(m) -5, {g® l)^(m) = ^S'M-
Let us again use the notation i{m) = J^ILi^* ® ^» while we write
^(^g{m)) = ^j^ifrij (gMj. Our second assumption can then be written
Yli^^dim) ®Si = Y^.^Jrij ®Sj,
It is now easy to prove that ^f is a morphism of descent data:
A (g) g)(l>{m (g) 5) = A (g) 5f) (Y^i^i^^ ® ^^^) = Yl'^-i^^ ® 9{i^is)
= YTi^i^^ ® 5^(^1M = EZj^i^i ® s'C^i)) A ® s)

246 Monads
Conversely, consider now a descent datum (/>: M^rS >S^rM over
an 5-module M. For simplicity, we shall write 0(m0l) = Yla=i^ct'S>ma'
The E(8)H5)-linearity of 0 means that, given 5,t G 5,
0(mt 0 5) = > Sat 0 ma5.
With this notation, the first condition in 4.7.1 for being a descent datum,
applied to an element m® 1, becomes
m-> Sarria,
while the pentagonal condition gives
(I0 0)@0l)(m0 1(8I)
= (lBH)@(m(8)lHl)
= A (g) 0) f ^ _ 5a ® ma 0 1 j
5a <S>(l>{ma 0 1),
Q=l
A 0 (JMs){(t> ^ 1)A <^ ^Ss)(m 0101)
= A0 crMs){(t> ® l)(m 0101)
= (l0c7Ms)@(m0lHl)
= A 0 aMs) i Xla=i^^ 0 ma 0 1 j
5a0l0ma,
Q=l
which finally yields
E5a 0 Mrria 01)= 7 5a 0 1 0 ma-
Q=l ^—^Q=l
Given such a descent datum 0 on an 5-module M, we define ^ to be
the composite
M—^^1^^—^M^rS ^—>S^rM ^^^ >M^rS
where riMijn) = m 0 1. With the previous notation
i{rn) = asM(t>{m 0 1) = (Tsm i X^^^^^a 0 ma j = y^^_^ma 0 Sa-
We shall prove that (M, 0 is a T-algebra.

4-7 A glance at descent theory 247
First of all, ^ is 5-linear. Indeed, given m G M, t G S^
^{mt) = asM(t>{mt (g) 1) = asM f ^ _ 5at (g) m^ J
= y^ rria ® Sat = I y^ rria ® «« ) t
The first condition for a T-coalgebra is just (see 4.1.2)
y^—'Q=l / ^—'Q=l
To prove the second condition for a T-coalgebra, observe first that the
pentagonal condition for a descent datum yields in particular, with the
previous notation,
Xla=l^^ ® ^(^a) = Xla=l^^ ® ma ® 1.
It is then easy to perform the various computations.
(^®m(m) = (^®l)
(Xla=l^^®^^
^(ma) ® 5a = > TTla ® 1 ® 5a
Q=l ^—^Q=l
= y] ,MM(ma®5a) =/iM f y^ ma 0 5a )
Thus (M, ^) is indeed a T-coalgebra.
Choose another 5-module M, a descent datum 0: M<S>rS >S<S>rM
and the corresponding T-coalgebra (M, ^). If ^f: (M, (/>) > (M, </>) is a
morphism of descent data, let us prove that g: (M,^) >(M,^) is a
morphism of T-coalgebra, which will finally yield a functor
r: Des(/) ^ModJ.
We keep the notation (f){m 0 1) = Z)f=i^a (g> ma and write in the
same way (j){g{m) 0 l) = X^^^^s^ 0 m^. The assumption on g thus
implies

248 Monads
This easily yields
^{g 0 l)(m) = ^{9{m) 0 1) = 2^ ^^m^ 0 5^
= {9®m){m).
Let us now check that the functors 0 and r are reciprocal equivalences
of categories (see 3.4.3, volume 1). Since the statement on the morphisms
is obvious, it suffices to consider the case of objects. Let us start with a
T-coalgebra (M,0; we put 0{M,^) = {M,(j)) and t{M,(I)) = (M,^); we
must prove the equality ^ = |. We use the notation already defined in
this proof.
En
._ Si ®mi,
?(m) = C7SM0(^ ® 1)
= ^(m).
The converse relation is completely analogous. Starting with a descent
datum (M, 0), we construct t{M, 0) = (M,^) and 0{M,^) = (M,0); we
must prove the equality (j) = (f). With previous notation, we have
0(m 01) = 2_^ _ 5a 0 TTla,
^—'Q=l
(j)(rn 01) = 2_^ _ 0 5a 0 TTla = 0(m 0 1),
from which
0(m 0 5) = 0((m 0 1)A 0 s)) = {(j){m 0 1)) A 0 s)
= D>{m 0 1)) A0 5) = 0((m 0 1)A 0 s))
= (t){m<S> s),
since 0 and (/> are E0i?5)-linear. This proves the stated equivalence.
Finally let us compute the composite of the comparison functor /^ of
4.7.2 with our functor r. This is the functor
tok: ModR ^►ModJ, iV h^ {N(S)rS,^n),

4-7 A glance at descent theory 249
where ^n- N(SirS >N(S)rS(S)rS is defined by
^N{n ®s) = (c7s,iV(8)flS o (t>N){n <S>s®l)
= n® 1 (8M
and an i?-linear mapping g: N >N is mapped to g<S>ls' Now observe
that the two canonical natural transformations of the adjunction /i H /*
are given by
aiv*. N >f*fi{N) = N(S)rS, ni-> n 0 1,
f^M' f\f*{M) = M(S>rS >M, m 0 5 i-> 7715,
for N G Modi? and M G Mods. Therefore
(To,.)(iV) = (/,(iV),/,(a^)),
which is precisely the action of the comparison functor described in 4.2.1.
The same remark holds for morphisms, since (r o n){g) = f\{g). □
To get a characterization of effective descent morphisms, let us recall
two classical definitions.
Definition 4.7.5 Let M be a module over a commutative ring R with
unit. M is fiat when the functor
—®rM: Modi? >Modi?
preserves monomorphisms.
Definition 4.7.6 Let R be a commutative ring with a unit. A
monomorphism i: A >B of R-modules is pure when, for every R-module
M, the canonical mapping
i 0 1: A(S>rM >B<®rM
is still a monomorphism.
One should probably emphasize the difference between purity and
flatness. In both cases one is interested in getting a monomorphism of
the type
i 0 1: A<®rM >B®rM.
When this is the case for all monomorphisms i then M is flat; i is pure
when this is the case for all modules M.
Lemma 4.7.7 Let f: R >S be a homomorphism of commutative
rings with unit. Iff is a pure monomorphism of R-modules, then
—®rS: fAodR >Mod5

250
A ^ A^rR
B ^ B(S)rR
Monads
1®/
-^A(SirS
1®/
Diagram 4.42
9^1
—>B®rS
/i(8)l
A) is a faitbful functor,
B) reflects isomorpbisms.
Proof Since Modi? and Mods are abelian (see 1.4.6.a), to prove that
—<S>rS reflects isomorphisms, it suffices to prove that it reflects both
monomorphisms and epimorphisms (see 1.5.1). And this will be the
case as long as —<S)rS is faithful (see 1.7.6). So consider two morphisms
g, h: A I B of i?-modules such that g®ls =" h®ls' Diagram 4.42 is
coromutative and, by assumption, the horizontal arrows are
monomorphisms. Therefore since
A (g) f)g = (^ (g) 1)A (g) /) = (/, (g) 1)A (g) /) = A (g) f)h
we deduce g = h. D
A classical theorem in descent theory is then:
Theorem 4.7.8 Let f: R >S be a bomomorpbism of coirmiutative
rings witb units. Suppose tbat:
A) f is a pure monomorpbism of R-modules;
B) S is a fiat R-module.
Under tbese conditions, f is an effective descent morpbism.
Proof With 4.7.4 in mind, it suffices to prove the comonadicity of the
functor -(S)rS: Mod^ >Mod5. To do it, we apply 4.4.4.
We know already that —<S)rS = f\ has a right adjoint /*; in
particular —<S>rS is right exact (see 3.2.1, volume 1, and 1.11.2, this
volume). By 3.9.3, the morphism /: S >R induces an algebraic functor
Mods >ModR, thus the functor —®rS: Mods ^-Mod/? preserves
monomorphisms since, by assumption, the composite
ModH -®R^y Mods > ModR
does, and the algebraic functor reflects limits (see 3.4.1). Thus —(SirS
is exact (see 1.11.4) and therefore preserves all equalizers (see 1.11.2).

^.1 A glance at descent theory
251
N y-
f
n
' ^ u
-> A
1®/
:; B
l(^g 1®/
u(8)l
1®^
K >—-—>A(SirSZIIZXB(SirS
Diagram 4.43
On the other hand Modi? has all equalizers (see 1.4.6.a). Therefore the
dual conditions of B) (a) and B)(c) in 4.4.4 are already satisfied and, by
4.7.7.B)(b), is satisfied as well. D
In 4.7.8, the purity of / is in fact a necessary condition for / being an
effective descent morphism (see exercise 4.8.12); the flatness of S is not
a necessary condition (see exercise 4.8.13).
Theorem 4.7.9 Let f: R >S be a homomorphism of commutative
rings with units. If f admits an R-Unear retraction, f is an effective
descent morphism.
Proof Again with 4.7.4 in mind, it suffices to prove the comonadicity of
the functor —<S>rS: Modi? ^-Mods. We know already that —<S>rS = f\
admits /* as a right adjoint.
If / admits a retraction g: S >R^ then from g o f = Ifi we deduce
(S' ® 1m) o (/ ® 1m) = li?(8)M for every i?-module M. In particular each
/ (8) 1m is a monomorphism and thus / is pure. By 4.7.7, this implies
that —(S)rS reflects isomorphisms.
It remains to check condition B)(c) of 4.4.4. Let us consider two mor-
phisms u^v: A ^B in Modi? whose images by —®rS have a split
equalizer. This yields diagram 4.43 in Modi? where k is the split
equalizer of (tx (g) 1, i; (g) 1), i.e.
{u<S>l)ok = {v(Sil)ok, 5ofc = l, r o (tx® 1) = 1, r o (v <S>1) = ko s.
If n is the equalizer of {u^v)^ the commutativity of the squares on the
right implies the existence of factorizations f,g through the equalizer,
with fc/ = A 0 f)n and ng = {1<S> g)k. It remains to put a = ^5A 0 /)

252 Monads
and p = A (g) g)r{l 0 /) to complete the diagram. If we prove that
the top line is a split equalizer, this split equalizer will be preserved by
every functor (see 2.10.2, volume 1) thus in particular by —<S>rS; this
will conclude the proof. And indeed
an = gs{l 0 f)n = gskf = gf = l^
pu = {l<S> g)r{l <S> f)u = (l® g)r{u ® 1)A ® f)
= A®^)A®/) = 1,
pv = {l® g)r{l (8) f)v = A (g) g)r{v (g) 1)A (g) /)
= A (g) g)ks{l <g) /) = ngs{l <S> f) = na. D
4.8 Exercises
4.8.1 Describe expUcitly the monad on Set admitting the category of
real vector spaces as category of algebras.
4.8.2 Let T = (T, e, /jl) be a monad over Set. Prove that the category Set^
of T-algebras is equivalent (in a bigger universe) to the full subcategory
of Fun(SetT,Set) whose objects are the product preserving functors.
4.8.3 Let ^ be a finitely complete category. Show that there is a
coincidence (up to equivalences of categories) between:
A) the localizations of ^ (see 3.5.5, volume 1);
B) the categories of T-algebras for the idempotent and left exact
monads T on ^ (i.e. the idempotent monads T = (T, £:,//)) such that T
preserves finite limits).
4.8.4 Let ^ be a complete and cocomplete category and T = (T, e, //) a
monad on ^ with finite rank. Prove that the category ^'^ of T-algebras
is cocomplete (compare with 4.3.6).
4.8.5 In a category ^, a pair of morphisms f^g: ff > A induces a
relation on each set ^(X, A) of morphisms: the relation constituted of
all the pairs {f ox^gox) for x: X >R. Prove that all those relations
are reflexive if and only if /, g are two epimorphisms with a common
section. Such a pair (/, g) is called a reflexive pair.
4.8.6 In the adjoint lifting theorem (see 4.5.5), prove that the two
morphisms ud and FL{$) admit FL{^d) as a common section. Conclude
that the assumption on s/ can be weakened by assuming just the
existence of coequaUzers of reflexive pairs (see 4.8.5).

4.8 Exercises 253
4.8.7 In the situation of 4.5.9, prove that the functor Q is full and
faithful if and only if each morphism Ac is an epimorphism in ^.
4.8.8 Let ^ be a well-powered category and T = (T, e, //) a monad on
^. Prove that the category ^^ of T-algebras is well-powered.
4.8.9 Let CompLat be the category of complete lattices and (AV)"
preserving maps. Prove that the corresponding underlying set functor
U: CompLat >Set is not monadic. [Hint: U does not have a left
adjoint, although it preserves limits; the "free complete lattice" on three
generators turns out already to be a proper class, not a set.]
4.8.10 Let T be a monad on the category of sets. Show that T has finite
rank iff the free algebra F{1) on the singleton has the following property:
for every set X and every morphism of T-algebras g: F(l) >F(X),
g factors through some F{i): F{Y) >F{X), with i: Y >X the
inclusion of a finite subset.
4.8.11 In the situation of 4.7.1, prove that given an {S(S)RS)-lmeai
mapping (/>: M^S^rS >S(S^rM satisfying the two given conditions, (f)
is necessarily an isomorphism and thus a descent datum.
4.8.12 A homomorphism /: R >S of commutative rings with unit
is called a descent morphism when the comparison functor defined in
4.7.2 Modi? >Des{f) is full and faithful. Prove that / is a descent
morphism if and only if / is a pure monomorphism in the category of
i?-modules.
4.8.13 Consider a homomorphism /: R >S of commutative rings
with unit. The mapping {^^): R >R x 5 is another homomorphism
of commutative rings with unit and it admits a retraction. Show that
n general, R x S is not a fiat i?-module. [Hint: choose R = Z and
S = Z/2Z.] Compare with 4.7.8 and 4.7.9.

5
Accessible categories
In chapter 11, we have studied those algebraic theories which can be
described via "finite powers": a theory was a category T with objects
T^, T^,..., T'^,... and a model was a functor T >Set preserving finite
powers.
Replacing "finite powers" by "finite limits" leads to the notion of a
locally finitely presentable category of models. A theory is now a small
category ^ with finite limits while a ^-model is a functor ^ >Set
preserving finite limits.
In section 6.3, volume 1, we observed that in the absence of finite
limits, the notion of a "flat functor" was a good substitute for the notion
of a functor preserving finite limits. Therefore a further generalization
consists in considering a theory as being just a small category ^, while
a ^-model is a flat functor ST >Set. This leads to what is called an
"accessible" category of models.
In fact, the accessible categories admit an elegant axiomatic
description: they are those categories with a "good" set of generators. And
locally presentable categories are just those accessible categories which
are cocomplete (or, equivalently, complete). In particular, the existence
of generators makes the accessible categories a fruitful context in which
to develop categorical constructions which, in general, require smallness
conditions.
Finally let us mention that if the categories of limit preserving
functors are locally presentable, the categories of limit-colimit preserving
functors turn out to be accessible. We present in fact a more flexible
approach in terms of general cones and cocones: what is called the theory
of sketches. A sketch is a small category provided with a set of
distinguished cones and a set of distinguished cocones; a model of this sketch
is a set-valued functor transforming the distinguished cones into limit
254

5.1 Presentable objects in a category 255
cones and the distinguished cocones into coUmit cocones. The categories
of models of sketches are exactly the accessible categories.
5.1 Presentable objects in a category
In section 3.8, we studied those models M of an algebraic theory T which
are finitely presentable. This means that M can be described via finitely
many generators and finitely many relations (see 3.8.1). But in 3.8.14
we proved this to be equivalent to the preservation of filtered colimits
by the representable functor
ModrCM,-): Modr >Set.
Clearly, such a definition generalizes easily to the case of an arbitrary
category. For the sake of generality, we give it with respect to an arbitrary
regular cardinal a (see 6.4) and not just with respect to Hq.
Definition 5.1.1 An object M £ J^ of a category M is a-presentable,
for some regular cardinal a, when the representable functor
Ji{M,-): M >Set
preserves a-filtered colimits. An object M is presentable when it is a-
presentable for some regular cardinal a.
Clearly, the study of a-presentable objects will be pertinent just when
Jf has a-filtered colimits. Trivially, one has
Proposition 5.1.2 Let Ji be a category and a < C two regular
cardinals. Every a-presentable object is also C-presentable.
Proof This is simply because every /3-filtered colimit is a-filtered. D
Here is a useful technical characterization of a-presentable objects.
Proposition 5.1.3 Let Jl be a category and a a regular cardinal.
An object M E Ji is a-presentable precisely when, for every a-filtered
colimit L = colimit/ Mi in Ji, the following two conditions are satisfied:
A) every morphism m: M >L factors through one of the canonical
morphisms sf. Mi >L of the colimit;
B) when two morphisms mi,m2: M >Mi are such that Si o mi =
Si o 7712, for some canonical morphism Si'. Mi >L of the colimit,
there exists a morphism Sij: Mi > Mj in the original diagram such
that Sij O TTli = Sij O 7712-

256 Accessible categories
Proof There exists in Set a canonical mapping
colim^(M, Mi) >Jf{M, L)
obtained as a factorization through the coUmit of the canonical cocone
^{M,Si): J/{M,Mi) ^J/{M,L).
The object M is a-presentable when this canonical mapping is bijective,
for every a-filtered colimit L = colimi^/ M^. The first condition expresses
the surjectivity of the mapping and the second condition expresses its
injectivity. D
Clearly, when the canonical morphisms Si'. Mi >L are monomor-
phisms, the second condition of 5.1.3 vanishes.
Proposition 5.1.4 Let M be a category and a a regular cardinal. In
JKy an a-colimit of a-presentable objects, if it exists, is again an a-
presentable object.
Proof Consider a functor F: / >./#, where ^ is a category with
strictly less than a morphisms and each object F{J) is a-presentable
in JK. Suppose the colimit of F exists. Given an a-filtered colimit L =
coliniig/ Mi in J(, one has by 2.9.5 and 6.4.5, volume 1,
^{colimF{J),cqlimMi) = limj^^Jf {F{ J), cqlim Mi)
^ limje/ colim^(F(J), Mi)
^ colimlimje/^(F(J),Mi)
^ colim Jf (colimF(J),Mi). D
iei ^ JG/ ^ ^
5.2 Locally presentable categories
We refer to section 4.5, volume 1, as far as generators are concerned.
Definition 5.2.1 A category M is locally a-presentable, for a regular
cardinal a, wiien:
A) Jl is cocomplete;
B) M has a set {Gi)i^i of strong generators;
C) each generator d is a-presentable.
A category is locally presentable when it is locally a-presentable for
aonw regular cardinal a. The locally i^o-P^^sentable categories are also
called locally Gnitely presentable.
Let us start with some examples.

5.2 Locally presentable categories 257
Examples 5.2.2
5.2.2.a If T is an algebraic theory as in 3.3.1, the corresponding
category Modr of models is locally finitely presentable. Indeed Modr is
cocomplete (see 3.4.5) and the finitely generated free models constitute
a family of dense, thus strong generators (see 3.8.10, this volume, 4.5.5
and 4.3.6, volume 1). They are finitely presentable by 3.8.14.
5.2.2.b Let ^ be a small category. The category Fun(^, Set) of set-
valued functors on ^ is locally finitely presentable. Indeed it is
cocomplete (see 2.15.4, volume 1) and the representable functors constitute
a family of dense, thus strong generators (see 4.5.17.b, 4.5.5 and 4.3.6,
volume 1). Given T e ^, the functor on Fun(^, Set) represented by
^(r, —) is just the evaluation at T (see the Yoneda lemma, 1.3.3,
volume 1), thus it preserves all colimits (see 2.15.1, volume 1) and in
particular filtered colimits.
5.2.2.C Let ^ be a small category with finite limits. The
corresponding category Lex(^, Set) of finite limit preserving functors ^ >Set is
locally finitely presentable. Indeed it is cocomplete (see 6.2.4) and the
representable functors constitute a family of dense, thus strong
generators (same argument as in example b, since the representable functors
are left exact). Again given T e ^^ the functor on Lex(^, Set)
represented by ^(r, —) is just the evaluation at T (see the Yoneda lemma
again), thus it preserves filtered colimits (see 6.2.2, volume 1).
5.2.2.d Following the lines of section 6.4, volume 1, example 5.2.2.c
generalizes immediately to the case of a regular cardinal a. If ^ is a small
category with a-limits, the category a-Lex(^, Set) of a-limit preserving
functors ^ >Set is locally a-presentable.
5.2.2.e The category Barii of real Banach spaces and linear
contractions is locally ^<i-presentable. We know that Barii is cocomplete (see
2.8.6, volume 1) and admits R as a strong generator (see 4.5.17.e,
volume 1). It remains to prove that Bani(R, —), i.e. the unit ball functor,
preserves ^<i-filtered colimits. Let us thus consider an ^<i-filtered
diagram in Barii and the corresponding colimit L = colimi^/ Bi computed
in the category of real vector spaces; since this colimit is filtered, it is
computed as in the category of sets (see 3.4.2, volume 1). Define a norm
on L by
\\l\\=mi{\\b\\\beBi,iGl, [b]=l}
for every element I E L, seen as an equivalence class of elements in
the Bi's. If ||Z|| = 0, for every n > 0, n G N, we find bn G Bi^ such

258 Accessible categories
that [bi^] = I and \\bi^\\ < ^, where [bi^] denotes the equivalence class
of bi^ in the colimit. By ^<i-filteredness, we get an index i such that
all the elements bi^ are mapped onto the same element b e Bi] then
\\b\\ < \\bi^\\ < ^ for every n, thus ||6|| = 0 and 6 = 0; this implies
I = [b] = 0. Prom this it follows easily that L has been provided with
a norm. The space L is complete for this norm. Indeed given a Cauchy
sequence {lk)keN in L, let us write h = [bk] with bk G Bk. Again by
i^i-filteredness we can choose all the ft^'s in a same Bi. Given e > 0, we
have ||/fc — lm\\ < ^ for k^m sufficiently large; this implies that fixing k
and m, \\bk — bm\\ < £ in some Bj and, again by ^<i-filteredness, we can
choose je independently oik^m. Now letting e run through the numbers
i^ n 7»^ 0, n G N, we get a corresponding family of j^'s; by ^<i-filteredness,
we can map the corresponding Bj^ into a unique Bj where the image of
the elements bk now constitute a Cauchy sequence. This yields a limit
in Bj and thus in L. It is now straightforward to verify that L is indeed
the colimit of the original diagram in Barii. Just by construction, the
open unit ball of L is the filtered colimit (computed in Set) of the open
unit balls of the B^'s. Now if ||/|| = 1 for some element / G L, write /
as I = lim(l — ^)l and combine the argument concerning the open unit
ball with a limit argument, again by ^<i-filteredness.
5.2.2.f The category Cat of small categories and functors is locally
fmitely presentable. Indeed, it is cocomplete (see 5.1.7) and the one-
arrow category 2 is a strong generator (see 4.5.17.h, volume 1). Now a
filtered colimit J5f = colimit/^i of small categories is easily computed:
as far as the sets of objects are concerned, |J5f | is just the colimit |J5f | =
cdlimi^j 1^1^ in Set; next if L, V G J5f, J5f (L, V) is the filtered colimit of
the sets 9Si{Li,Vi) where L = [Li] and L' = [L^]. Since giving a functor
2 >^ is just giving an arrow of ^, the construction of the colimit J5f
indicates that CatB, —) indeed preserves filtered colimits.
The aim of this section is to prove that every locally a-presentable
category Jt is of the type a-Lex(^,Set), for some small category ^
with a-limits. But let us first observe the following.
Proposition 5.2.3 Let a < /3 be regular cardinals. Every locally a-
presentable category is also locally ^-presentable.
Proof Follows from 5.1.2. D
Lemma 5.2.4 Let a be a regular cardinal and Jl a locally a-presentable
category. With the notation of 5.2.1, write ^ for the full subcategory of

5.2 Locally presentable categories 259
Ji generated hy the generators (Gi)i^i. The full closure ^ of ^ in M
under a-colimits exists and has the following properties:
A) SP is equivalent to a small category;
B) SP has a-colimits computed as in Jt\
C) every object in SP is a-presentable in Jt.
Proof It suffices to define ^ by transfinite induction:
• ^0 = ^;
• ^^+1 is the full subcategory of Ji whose objects can be obtained as
colimit of a functor f >^^5 with f a category with strictly less
than a arrows;
• ^7 = U^<7^i9 fo^ ^ li^i^ ordinal 7.
Then ^ is defined to be ^a-
Let us prove conditions A) and C) of the statement by induction on
the ordinal /?.
• ^ is small since / is a set and by assumption, each object Gi is
a-presentable;
• up to equivalence, there is just a set of categories / with strictly less
than a arrows, thus just a set of functors F: f >^^, assuming
^p to be small; this implies the smallness of ^^+1; condition C)
holds for ^^+1 just by 5.1.4;
• the case of a limit ordinal is obvious.
Let us prove condition B) of the statement. For a functor F: / >^a
where f has strictly less than a arrows, each arrow F{j) is in some ^aj
for aj < a. By regularity of a, there exists C < a such that aj < C for
each j. Therefore the colimit of F is already in ^^+1, still with /?+1 < a.
Observe that every a-cocomplete replete subcategory J, ^ C J C ^,
contains ^0 and, by transfinite induction, every ^^. So ^ is indeed the
closure of ^ under a-colimits. D
Lemma 5.2.5 Under the conditions of lemma 5.2.4:
A) for every object M G Jf, the category ^/M is a-filtered;
B) the objects of ^ constitute in M a dense family of generators.
Proof The objects of ^/M are thus the pairs (P,p) with P e ^ and
p: P >M; an arrow /: (P,p) >{Q^q) is a morphism /: P >Q
such that qo f = p. We write T: ^/M >^ for the functor applying
(P,p) on P and f on f (see 4.5.4, volume 1). The category ^/M is

260 Accessible categories
certainly a-filtered since ^ has a-colimits (see 5.2.4). We must prove
that the colimit of T is just (p: P >M)(p^p)^^/M-
Let us consider the colimit [L^S(p^p)).p )^^/m ^^ ^ ^^ '^' Since the
morphisms p: r(P,p) >M constitute a cocone on F, we get a unique
factorization A: L >M such that Ao5(Pp) = p for every (P,p) G ^/M.
We must prove that A is an isomorphism.
We consider first diagram 5.1 where the coproduct is indexed by
all the pairs (Gi,p) with i e I, Gi e ^ and p: Gi >M. Writing
cr^Gi.p)'- Gi > Y[Gi for the canonical morphisms of the coproduct, u
is the unique morphism such that u o (T(Gi,p) = P ^^^ ^ is the unique
morphism such that v o (T(Gi,p) = ^(Gi.p)- In particular,
A o^ oa(Gi,p) = Ao S(Gi,p) =p = uo a(G,,p),
which yields X o v = u. By assumption, t^ is a strong epimorphism (see
4.5.3, volume 1), thus A is a strong epimorphism as well (see 4.3.6,
volume 1). So by 4.3.6, volume 1, again, it remains to prove that A is a
monomorphism.
Let us consider two morphisms x^y: X ^ Tj in Ji such that A o x =
X o y. Since the G^'s constitute a family of generators, it suffices to
prove that for each i G / and each z: Gi >X, xo z = yo z. But the
coUmit {L^S(^p^p)).p )^^/^ is a-filtered, since ^/M is a-filtered, and
Gi is a-presentable. Therefore xoz and yoz factor through some term
of the colimit (see 5.1.3) and, by filteredness of the colimit, there is no
restriction in supposing it is the same term. So let us fix (P,p) G ^/M
and x', y'\ Gi >P such that 5(p,p) ox^ = xo z and S(p^p) oy' = yo z.
This yields
p o x' = A o S(^p^p) ox^ = Xoxo z^ poy' = Xo S(^p^p) oy' = Xoyo z^
from which we get two morphisms in ^/M,
x': {Gi.Xoxoz) >(P,p), y': {GuXoyoz) >(P,p).

5.2 Locally presentable categories 261
Therefore, since Xox = Xoy,
xoz = 5(p,p) ox' = S(GiMz) = ^(GiMz) = ^(P,p) oy' = yoz. D
Lemma 5.2.6 Under the conditions of 5.2.4, SP is the full subcategory
of a-presentable objects in Jt.
Proof By 5.2.4, it remains to prove that every a-presentable object
M G ./# is in ^. Using the notation of 5.2.5, we write M as a filtered
colimit M = colim(pp) P where (P,p) runs through ^/M. By 5.1.4,
the identity 1m- M >M factors through some term of the colimit,
yielding (P,p) G ^/M and m: M >P such that pom = 1m- Thus M
is a retract of P, i.e. (see 6.5.4) the coequalizer of the two morphisms
Ip, mop: P^^,P. Since by 5.2.4 ^ is stable under a-colimits, M itself
is in ^. D
Theorem 5.2.7 Let ^ be a locally a-presentable category, for some
regular cardinal a. There exists a small a-complete category 3' such
that Jl is equivalent to the category a-Lex(^,Set) of set-valued a-left-
exact functors. The dual of ST is itself equivalent to the full subcategory
of Ji generated by the a-presentable objects.
Proof We use the notations of the previous lemmas and, by 5.2.4,
choose a set Q of objects of J^ such that each object of ^ is isomorphic
to some object of Q. We write J for the full subcategory of Jf generated
by the elements of Q; ^ is defined as the dual of J. We know by 5.2.5
that the objets of J constitute a dense family of generators in J^, thus
the functor
Y:J^ >Fun(J*,Set), M^Jf{-,M)
is full and faithful (see 4.5.14, volume 1). But each representable
functor ./#(—, M): J^ >Set transforms colimits into limits (see 2.9.5,
volume 1); since ^ is stable in J^ for a-colimits (see 5.2.4), each functor
J({—,M): 2L >Set transforms a-colimits into a-limits. Thus we have
already defined a full and faithful functor
Y'.M >a-LexE",Set), Mh-^^(-,M).
To get an equivalence of categories, it remains to prove that every a-
left-exact functor F\ ^ >Set is isomorphic to Y(M^, for some object
M e M, But Y preserves a-filtered colimits since, given a a-filtered
colimit L = colimj^J Mj in Ji,

262 Accessible categories
Y{L){Q) = J^{Q,L) = Jf{Q,colimMj)
= colim J^iQ.Mj) = co\imY{Mj){Q)
jeJ jeJ
just because each Q e lis a-presentable; thus Y{L) = colim^^ jy(Mj)
since a-filtered colimits are computed pointwise in a-Lex(^,Set) (see
6.4.9, volume 1). Now an a-left-exact functor F: 3' >Set is the a-
filtered colimit F\ colim(Tt) ^^->-)-> where (r,t) runs through the
category of elements of F (see 054.7). Considering the composite
Elts(F)* >5"* ^21A M,
we get a functor defined on a small a-filtered category; write (L, 5(T^t))
for its colimit. We have
y(L) ^ y(colimT) ^ colimy(T) ^ colim^(-,r)
^ colim J(-,r) ^ colim^(r, -)
(T,t) (T,t)
= F
since T itself is an object of J. D
Corollary 5.2.8 Let a be a regular cardinal In a locally a-presentable
category:
A) small limits and colimits exist;
B) a-limits commute with a-filtered colimits;
C) a-61tered colimits are universal
Proof By 5.2.7 and 6.4.9, volume 1, conditions A), B) and C) hold.
D
Corollary 5.2.9 Let a be a regular cardinal and ST an a-complete
category. In the category a-Lex(^, Set) of a-left-exact functors, the a-
presentable objects are exactly the representable functors.
Proof The representable functors constitute a family of strong (and
even dense) generators (see 5.2.2.d); this family is closed under a-colimits
since the Yoneda embedding
Y: Sr >a-Lex(.^, Set), T ^ ^(T, -)
transforms a-limits into a-colimits (see 6.4.9, volume 1). Therefore the
"minimality" of ^ in lemma 5.2.4 yields ^ = ^ = Y{3r)\ thus by 5.2.6
every a-presentable object is a representable functor. D

5.3 Accessible categories 263
Proposition 5.2.10 Let Ji hea locally presentable category. For every
object M e J^ there exists a regular cardinal aM such that M is aM-
presentable.
Proof Suppose J^ is locally a-presentable; we use the notation of 5.2.7.
Since J is small, the category J/M is small; let us choose a regular
cardinal aM larger than a and such that J/M has strictly less than
aM arrows. By 5.1.2, each object Q £ ^ is a-presentable, thus aM-
presentable. So M is a aM-colimit of aM-presentable objects (see 5.2.4),
thus M is aM-presentable (see 5.1.4). D
5.3 Accessible categories
Bearing in mind lemma 5.2.5, we make the following definition:
Definition 5.3.1 A category Jt is a-accessible, for some regular
cardinal a, when:
A) M has a-filtered colimits;
B) Jt has a set (Giji^i of dense generators;
C) writing ^ for the full subcategory of Jt generated by the generators
Gi, the category ^/M is a-filtered for every object M G Jt;
D) each generator Gi is a-presentable.
A category is accessible if it is a-accessible for some regular cardinal a.
Clearly, various equivalent formulations of definition 5.3.1 could be
given. Here is a useful "weakening" of it.
Proposition 5.3.2 A category Jt is a-accessible, for some regular
cardinal a, if and only if:
A) Jt has a-filtered colimits;
B) there exists a set {Gi)i^i of a-presentable objects such that each
object M £ Jt is the colimit of some a-filtered diagram in the full
subcategory ^ generated by the Gi's.
Proof Obviously, the conditions of 5.3.1 imply those of 5.3.2.
Conversely, consider a functor F: / >^, with / a small a-filtered
category and (M, sj)j^^ the colimit of F in Jt. Let us prove that ^/M is
a-filtered.
By 5.1.3, every arrow /: Gi >M, with z G /, factors through some
sj: F{J) >M, J e /. Thus, given a a-family {Gk.fk) of objects of
^/M, the a-filteredness of / allows us to factor all the /^'s through
the same sj: F{J) >M, yielding morphisms {Gk^ fk) >{F{J)^sj)

264 Accessible categories
in ^/M. In an analogous way if gk'- (G^f) >(G'^f') is an a-family
of morphisHis in ^/M, let us factor /' through some sj via a mor-
phism g\ G' >F{J). By the second condition of proposition 5.1.3,
two morphisms g o gj^^g o g^: G >F{J) can be coequalized by some
F{j): F{J) >F{J)\ just because sj o gj^ = f — sj o gy. Using once
more the a-filteredness of f^ we can even choose a single morphism j
having the required property for all pairs {k^k'). But this finally yields
a morphism F{j) o g: (G'.f) >{F{J'),sj>) in ^/M coequalizing all
the morphisms gk- This proves the a-filteredness of ^/M, thus condition
5.3.1.C). But our argument proves also that [F{J)^sj)^ ^ is cofinal in
^/M (see 2.11.2, volume 1), implying condition 5.3.1.B). D
As a first example we get
Proposition 5.3.3 An accessible category is locally presentable if and
only if it is cocomplete.
Proof A locally presentable category is cocomplete by definition and
accessible by 5.2.5 and 5.2.4. Conversely a cocomplete accessible
category is locally presentable because a family of dense generators is also
a family of regular (see 4.5.5, volume 1) and thus strong (see 4.3.6,
volume 1) generators. D
Thus the two notions of locally presentable and accessible category
differ essentially in the amount of colimits whose existence is required.
The fundamental example of an accessible category is described in the
following proposition.
Proposition 5.3.4 Let ^ be a small category and a a regular cardinal.
The category a-Flat(^, Set) ofa-flat functors^ >Set is a-accessible.
Proof By 6.4.14, volume 1, a-Flat(^,Set) has a-filtered colimits
computed pointwise. By definition of an a-flat functor F: ^ >Set (see
6.4.10, volume 1), its category of elements is a-cofiltered. By 4.5.17.b,
volume 1, F is thus the a-filtered colimit of the representable
functors above it, in Fun(^, Set). Since the representable functors are a-flat
(see 6.4.12, volume 1) and a-filtered colimits are computed pointwise in
the categories Fun(.^,Set) and a-Flat(.^, Set) (see 2.15.2 and 6.4.14 of
volume 1), the representable functors finally constitute a set of dense
generators for a-Flat(.^,Set); as we observed, condition C) of 5.3.1 is
satisfied as well. By the Yoneda lemma the functor on a-Flat(.^,Set)
represented by a representable functor .^(T, —) is just the evaluation at
T; this preserves a-filtered colimits since those are computed pointwise.
So the representable functors are a-presentable in a-Flat(.^, Set). D

5.3 Accessible categories 265
The example of a-flat functors is in fact generic, as attested by the
following theorem.
Theorem 5.3.5 Let a he a regular cardinal and M an a-accessible
category. There exists a small category 3' such that M is equivalent to
the category a-Flat(^, Set) ofa-Qat functors ^ >Set.
Proof We use the notation of 5.3.1 and define ^ as the dual ^ = ^*
of ^. Since the G^'s constitute a family of dense generators, the functor
Y:J^ >Fun(^*;Set), M^J^{-,M)
is full and faithful (see 4.5.14, volume 1). Now the category of elements
of the functor ./#(—, M): ^ >Set, by the Yoneda lemma, is equivalent
to the category ^/M; thus it is a-filtered by assumption. This proves
that each functor ./#(—,M): ^* >Set is a-flat (see 6.4.10, volume 1)
so that we already have a full and faithful functor
Y: Ji >a-FlatE", Set), M ^ Ji{-, M).
To get an equivalence of categories, it remains to prove that every
a-flat functor F: ^ >Set is isomorphic to Y{M), for some M G J^.
But the Yoneda embedding preserves a-filtered colimits since, given an
a-filtered colimit L = colim^^j Mj in Jf and an object Gi G ^,
Y{L){Gi) = J^{Gi,L) = Jf{Gi,cqlimMj)
= cqlim^{Gi,Mj) = colimy(Mj)(Gi),
just because each Gi G ^ is a-presentable; thus Y{L) = colim^^j Y{Mj)
since a-filtered colimits are computed pointwise in a-Flat(^, Set); see
6.4.14, volume 1. Now an a-flat functor F: ^ >Set can be written as
an a-filtered colimit F = colim(xt) ^(T^—)^ where (T^t) runs through
the category of elements of F; see 6.4.13, volume 1. Considering the
composite
Elts(F)* >5"* ^ ^ C ^,
we get a functor defined on a small a-filtered category; write (L, S(xt))
for its colimit. We have
Y{L) ^ y(colimr) ^ colimy(r) ^ colim.^(-,r)
^ colim^(-,r) ^ colim.^(r, -)
since T itself is an object of ^. D

266 Accessible categories
Proposition 5.3.6 An accessible category is Cauchy complete.
Proof Consider the category ^ with one single object *, the identity
arrow on * and an idempotent arrow e: * >*, thus e o e = e. Since
eol^ = eoe, it follows immediately that ^ is a-filtered for every regular
cardinal a. Therefore, given an accessible category J^ and a functor
F: ^ >^, the colimit of F always exists. This implies the splitting
of idempotents in Ji\ see 6.5.4, volume 1. D
Proposition 5.3.7 Let Ji be an accessible category. For every
object M G ^j there exists a regular cardinal aM such that M is aM-
presentable.
Proof Suppose J^ is a-accessible; we use the notation of 5.3.1. For
M G ./#, the category ^/M is small; let us choose a regular cardinal
aM larger than a and such that ^/M has strictly less than aM arrows.
By 5.1.2, each object G^ G ^ is aM-presentable. So M is a aM-colimit
of aM-presentable objects, since the G^'s constitute a family of dense
generators; thus M is aM-presentable (see 5.1.4). D
Proposition 5.3.8 Let Ji be a a-accessible category. The full
subcategory of a-presentable objects is equivalent to a small category.
Proof We use the notation of 5.3.1. Let M G ./# be a a-presentable
object; by definition of a-accessibility, we can write M as a small a-filtered
colimit M = colim(^G,g)e^/M G. The identity arrow 1m- M >M
factors through some term of this colimit, by a-presentability of M (see
5.1.3); this yields an index (G,g) G ^/M and an arrow /: M >G
such that go f = l^. By 6.5.4, volume 1, M is the equalizer of the pair
of morphisms 1m, f ^ 9- G ^G. Since ^ is small, there is just a set
of such pairs and thus (up to isomorphisms) just a set of a-presentable
objects. n
Proposition 5.3.9 Let Ji bea a-accessible category. With the notation
of 5.3.1, the family of functors
Ji{Gi,-): M >Set
collectively reflects isomorphisms (see 4.5.7, volume 1).
Proof Given /: A >B in ./#, suppose that each Ji{Gi,f) is an
isomorphism. Considering the full and faithful embedding of 4.5.14,
volume 1,
T:J( >Fun(^*,Set), Mh->^(-,M),

5.4 Raising the degree of accessibility 267
we have that Ji(Gi^ f) = T{f)Gi is an isomorphism for each Gi G ^.
Thus r(/) is an isomorphism, as is /, since T is full and faithful. D
Proposition 5.3.9 could be interpreted as the fact that the objects
Gi e^ constitute a family of strong generators (see 4.5.13, volume 1).
5.4 Raising the degree of accessibility
When a problem involves several accessible categories, it is clearly useful
(and often essential) to be able to consider them all as a-accessible, for
the same regular cardinal a. In the case of locally presentable categories,
this can be achieved in an obvious way: given a family {ai)i^i of regular
cardinals, just choose a regular cardinal a bigger than all the a^; by
5.2.3, every locally ai-presentable category is also a-presentable.
The case of accessible categories is more subtle. li a < /3 are regular
cardinals, every /3-filtered colimit is certainly a-filtered and every a-
presentable object is /3-presentable (see 5.1.2). Therefore, considering
the various conditions of definition 5.3.1, satisfied for a fixed regular
cardinal a:
• condition A) holds for every regular cardinal /3 > a;
• condition B) is independent of a;
• condition C) holds for every regular cardinal C < a;
• condition D) holds for every regular cardinal C > a.
Thus the obvious way to modify the degree of accessibility varies in
opposite directions for conditions A) and D) and for condition C).
The aim of this section is to show that nevertheless the degree of
accessibility of a category can always be increased to arbitrarily large
but well chosen regular cardinals. This is essentially a game on cardinal
arithmetic, which leads to the final conclusion expressed in theorem 5.4.5
and corollary 5.4.8.
Definition 5.4.1 Let a, C be regular cardinals. The cardinal a is sharply
less than C (which we write a</3) when a < C and, for every set X of
cardinality strictly less than /?, there exists a cofinal subset of cardinality
strictly less than C in the poset of those subsets of X with cardinality
strictly less than a.
We shall write Va{X) for the poset of those subsets of X with
cardinality strictly less than a. The condition of definition 5.4.1 means thus

268 Accessible categories
the existence of a subset Q C Va{X)^ with the cardinality of Q strictly
less than /?, and
yu eVa{X) 3V eQ Ucv.
We shall prove that when a is sharply less than /?, every a-accessible
category is also /3-accessible.
Lemma 5.4.2 Let a< C be two regular cardinals. For every category
^ with strictly less than C arrows, there exists a subset of cardinality
strictly less than C in the poset of those subcategories of^ with strictly
less than a arrows.
Proof Every infinite set X has the same cardinal as its set of finite
subsets. Since the subcategory ^ generated by a set X of arrows is
given by all the finite composites of arrows in X, it follows that if X has
cardinality strictly less than a, so does the subcategory ^ generated by
X.
Under the conditions of the statement, let us choose Q, a cofinal subset
of cardinality strictly less than C in the poset 'Pa(Ar(^)), where Ar(^)
is the set of arrows of ^. Each subcategory of ^ with strictly less than a
arrows is contained in some X £ Q, thus in the subcategory 2f generated
by X, which still has strictly less than a arrows. Thus the set Q of
the subcategories ^ generated by the elements X e Q satisfies the
conditions of the statement. D
Lemma 5.4.3 Let a</3 be two regular cardinals. Ifs/ is an a-filtered
category every subcategory ^ C s/ with cardinality strictly less than
/3 is contained in an a-filtered subcategory ^ whose cardinality is still
strictly less than C.
Proof We shall construct ^ by transfinite induction.
• ^0 = ^, thus#^o </?.
• Suppose that ^x is defined, for some ordinal A, with the condition
#^A < C- Since a < /?, we choose a cofinal subset Q, with #si < /?,
in the poset Cata(^A) of those subcategories of ^\ with strictly
less than a arrows (see 5.4.2). Since j^ is a-filtered, for every Sf G
Q we can choose a cocone {sx- X >A^)xesf on ^ in j^; see
6.4.4, volume 1. We define ^a+i as the subcategory generated by
^x and all the morphisms sx, for all X G ^ and Sf e Q. Since the
cardinality of Q and of each ^ is strictly less than /3, #^a+i < /?
by regularity of /3.

5.4 Raising the degree of accessibility 269
• If // < /? is a limit ordinal, we define ^^ = [Jx^fJ^x'^ by
regularity of /?, #^fj, < C. Putting ^ = ^^, we have thus constructed a
subcategory containing ^ and with cardinality strictly less than /3.
To prove that ^ is a-filtered, let us consider a category / with strictly
less than a morphisms and a functor F: / >^. For each arrow j G ^,
Fj is in some ^x^ with A < /?. By regularity of /?, all arrows Fj are in
some ^A5 for a unique A, since #/? < a < C. Therefore there is a cocone
on F in ^a+i? thus in ^; see 6.4.4, volume 1. D
Lemma 5.4.4 Let a</3 be two regular cardinals. Ifs/ is an a-Gltered
category, the poset ofa-Gltered subcategories ofs/ with cardinality strictly
less than C is /3-Gltered.
Proof If {^i)iei is a family of a-filtered subcategories of s/, with
#/ < C and for every i £ I^ #^i < /?, then by regularity of C the
subcategory ^ generated by the union of the ^^'s still has cardinality
strictly less than C. One concludes the proof by 5.4.3. D
Theorem 5.4.5 Let a<l3 be two regular cardinals. Every a-accessible
category is also C-accessible.
Proof We use the notation of 5.3.1. Certainly Ji has /3-filtered colimits
since it has a-filtered colimits.
Let us now consider all the objects H £ M which can be written as
the colimit of a functor
3C C ^jE ^—>Ji
where dC is an a-filtered subcategory of cardinality strictly less than /?
and r is the usual forgetful functor; we write ^ for the corresponding
full subcategory of Ji. By 5.L4, every object if G Jf' is /3-presentable.
But since ^ is a dense family of generators, each object if G Jf' is in
fact the colimit of the forgetful functor ^/if >./#; see 4.5.4, volume \.
Therefore if is completely determined (up to isomorphism) by a diagram
of objects Gi whose cardinality is strictly less than /?. Since ^ is small,
this implies that 2^ itself is equivalent to a small category. Write Jf for
a small full subcategory of ^, equivalent to 2^ and containing ^. We
shall prove that Jf is a subcategory exhibiting the /3-accessibility of M,
via lemma 5.3.2.
Given an object M G Ji., consider the situation
^/M C jr/M ^ >Ji

270 Accessible categories
where T is the usual forgetful functor (see 3.5.4, volume 1). Let us
consider a cocone tf^K.k)'- r(A', k) >N on F; its restriction to ^/M yields
a unique factorization m: M >N such that mok = t^K.k)-) for every
K e^^ just because ^ is a family of dense generators. To get the relation
mok = t(K,k) for every K G JT, it suffices to prove mo kol = t(^K,k) ^ I
for every i E I and every /: Gi >K\ and indeed
mokol = t(Gi,koi) == ^(K,k) o I
since /: (G^, kol) > (K, k) is a morphism of JT/M. Thus JT is a family
of dense generators which, as already observed, are /3-presentable.
Let us prove now that every object M £ J^ is a. /3-filtered colimit
of objects in Jf. ^/M is a-filtered by assumption. For every a-filtered
subcategory ^ C ^/M with #^ < /?, the colimit of the functor
^ C ^/M £ ^Ji
is (up to an isomorphism) some object K^c of JT. The various inclusions
^ C <3^ C ^/M yield corresponding factorizations
Ksc ^K^ ^^—^M
in Ji. By 5.4.4, we obtain in this way a /3-filtered diagram in JT together
with a cocone (s^: X^ >M)^ on this diagram. This cocone is in fact a
colimit one, just by the associativity argument of 2.5.7, volume 1, where
"finite" is now replaced by "strictly less than a". Thus M is indeed the
colimit of a /3-filtered diagram in JT. D
So theorem 5.4.5 shows that < is a good condition for increasing the
degree of accessibility. One can even prove that for two regular cardinals
a,/3, a</3 iff every a-accessible category is /3-accessible (see 5.7.10). But
we shall not need this result.
Now the problem which remains is to prove that given a regular
cardinal a, or even a family {oLi)i^i of regular cardinals, it is always possible
to find a regular cardinal C such that a</3, or ai</3 for every z G /. This
is just a small game on cardinal arithmetic, at some level which requires
of course the axiom of choice. Let us recall some classical notation and
results which can be found in most books on set theory and cardinal
arithmetic. All Greek letters denote cardinals.
• a^ stands for the successor cardinal of a;
• for a infinite, a^ is always a regular cardinal;
• a^^ stands for Y^^^^oC;
• if iS > 2, a < a<^;

)ri if 7 > /3;
5.4 Raising the degree of accessibility 271
• if /? is a regular cardinal,
• if X is a set with infinite cardinal a and C < a,
a^ = #{y|FCX, #y = /3};
• if X is a set with infinite cardinal a and /3 < a^
*Vp{x) = #{y| F c X, #F < /?} = E^^^"'' = "^^^'^
• an infinite set has the same cardinality as its set of finite subsets,
i.e. a'^^o = a for every infinite cardinal a;
• if ai < f3i for every i G /, then Y^i^jOLi < Hi^j/Si]
• ii a < /3 with /? infinite, a-\- C = C\
• if a is infinite, a = a^.
The notation #Z stands clearly for the cardinality of the set Z.
Lemma 5.4.6 Let a,/3. A,// be cardinals with a^/3 regular. If a < C
and for every X < a, /jl < C one has /jl^ < /?, then a<l3.
Proof Consider a set X with cardinality /jl < /3. One has #Pq;(X) =
fji^^; but since /3 is regular, our assumption implies fx^^ = Yl\<af^^ ^ f^-
So ^a(X) itself has cardinality strictly less than C and, certainly, is
cofinal in itself. D
Proposition 5.4.7 Let f3 be a regular cardinal. There exists a regular
cardinal f3 > C such that, for every regular cardinal a < C, one has a<C.
Proof Let us put /? = B*^^)"^; as a successor, this is a regular cardinal.
We use lemma 5.4.6 to prove that a < C implies a < /?, for a regular
cardinal a. We choose cardinals A < a and //</?; we must check that
^^ < ^. Since /jl < B<'^)+, we have // < 2^^ and thus
/<B<^)^ = 2<^<B<^)+=;9
since X < a < C. D
Corollary 5.4.8 Let {ai)i^i be a family of regular cardinals. There
exists a regular cardinal P such that, for every i e I, ai</3.
Proof Choose a regular cardinal 7 such that a^ < 7, for every i G /.
Put /3 = 7 and conclude the proof by 5.4.7. D

272 Accessible categories
5.5 Functors with rank
We shall now study those functors which are compatible with the notion
of accessibility.
Definition 5.5.1 A functor F: s/ >^ has rank a, for some regular
cardinal a, when F preserves a-filtered colimits. It has rank when it has
rank a for some regular cardinal a.
Clearly, the study of functors with rank a can be interesting only in
the case where a-filtered colimits exist in the corresponding categories.
Proposition 5.5.2 Let a < C be regular cardinals. Every functor with
rank a has also rank C.
Proof Every /3-filtered colimit is a-filtered. D
The following propositions characterize functors with rank in the case
of accessible categories.
Proposition 5.5.3 Let s/ be an accessible category. A set-valued
functor F: si >Set has rank if and only if it is a small colimit of repre-
sentable functors.
Proof Suppose first that we can write F = coliirii^i si{Ai^ —) where
the corresponding diagram is small. Each object Ai has some rank a^;
choosing a regular cardinal a such that a > ai for each i, 5.1.2 implies
that each Ai is a-presentable. Then each functor si{Ai^—) preserves
a-filtered colimits, and thus so does the colimit F by 2.12.1, volume 1:
F (colim Bj) = colim j^ (Ai, colim Bj)
^ jeJ -^^ iei ^ jeJ -^^
= coUm colim si(Ai, Bj)
iei jeJ '''
= colim colim si{Ai, Bj)
jeJ iei
^ colimF(R)
for an a-filtered colimit colim^^j Bj. Thus F has rank a.
Conversely suppose F has rank C and si is 7-accessible. By 5.5.2, 5.4.8
and 4.4.5, volume 1, we can choose a regular cardinal a such that F has
rank a and si is a-accessible. Using the notation of 5.3.1, we consider
the restriction of F to ^ and the corresponding category of elements,
Elts(F o i) ^—> ^ > i > s^ ^—> Set.
Writing Y for the contravariant Yoneda embedding,
Elts(Foz) ^—><S ^—>Fun(j/,Set),

5.5 Functors with rank 273
we can consider the colimit of this composite, since Elts(F o i) is small,
due to the smallness of ^. By the Yoneda lemma, an object (G, g) G
Elts(F o i) determines a corresponding natural transformation g:
si(G^ —) => F\ this yields a factorization
(z?: colimja^(G,-) => F
and it suffices to prove it is an isomorphism. Each functor si(G^ —)
preserves a-filtered colimits since G is a-presentable, thus the colimit
also preserves a-filtered colimits (by same argument as in the first part
of the proof); on the other hand F preserves a-filtered coUmits. By a-
accessibility of si every object A^ si can be presented as the a-filtered
colimit of the functor ^ jA >si, Since both colim(G p) si(G^ —) and F
preserve this a-filtered colimit, the isomorphism colim(Gp) J2/(G, A) =
F{A) will hold as soon as the isomorphism colim((^ p) si(G, G) = F{G')
holds for every object G' G ^. But this is equivalent to
{Foi)(G')= colim "S^G.G')
(G,p)GEItS(Foi)
which is precisely the content of 2.15.6, volume 1. D
Proposition 5.5.4 Let F: si >^ be a functor between accessible
categories si ^0^, Then F has rank iff for every object B e ^, the
composite functor ^(B, F—): si >Set has rank.
Proof Suppose F has rank a and fix B G ^; from 5.3.6 B is /?-
presentable for some regular cardinal /?. By 5.5.2 and 5.1.2, we choose a
regular cardinal 7 such that F has rank 7 and B is 7-presentable. Then
both functors F and ^(B, —) preserve 7-filtered colimits and thus so
does the composite ^{B,F-).
Conversely, by 5.4.5 and 5.4.8 choose a regular cardinal a such that
si and ^ are a-accessible. Consider a family Bi e ^,i e I, of objects of
^ exhibiting the a-accessibility of ^, as in 5.3.1. By assumption, each
composite ^{Bi^F) preserves a^-filtered colimits for some regular
cardinal ai. Choose C a regular cardinal bigger than a and each a^; then
si and ^ have /3-filtered colimits and the functors ^{Bi, -), ^{Bi, F-)
preserve them. To conclude that F preserves /3-filtered colimits, it
remains to show that the family ^{Bi, -) of functors reflects /3-filtered
colimits or, equivalently, by the argument of 2.9.7, volume 1, reflects
isomorphisms. This is the case by 5.3.8. D
Proposition 5.5.3, together with 5.5.4, shows that a functor between
accessible categories has rank precisely when it satisfies good smallness

274 Accessible categories
conditions. Here is another striking smallness condition satisfied by those
functors (see 3.3.2, volume 1).
Proposition 5.5.5 Every functor with rank F: si >^ between
accessible categories satisfies the solution set condition at every object
B € ^; see 3.3.2, volume 1.
Proof Fix B G ^, which is /3-presentable for some regular cardinal C
(see 5.3.6). By 5.1.2, 5.5.2, 5.4.5 and 5.4.8 choose a regular cardinal a
such that B is a-presentable, s/,^ are a-accessible and F has rank a.
With the notation of 5.3.1, write ^ for the small full subcategory of s/
exhibiting the a-accessibility oi s/ as in 5.3.1.
Given A£ j^ and b: B >FA, write A = colimi^/ Gi where Gi^^
and the colimit is a-filtered (see 5.3.1). By choice of the cardinal a,
F{A) = colimit/ F{Gi) so that the morphism b factors through some
F{si): F{Gi) >F{A)^ where Si is a canonical morphism of the colimit
(see 5.1.3). Thus ^ is a solution set for B. D
With 3.3.3, volume 1, in mind, the next result is in a way related to
the previous one.
Proposition 5.5.6 If a functor F: si >^ between accessible
categories has a left adjoint, it has rank.
Proof Choose first a regular cardinal a such that both j^ and ^ are
a-accessible (see 5.4.5 and 5.4.8). Using the notation of 5.3.1, write ^
for the small full subcategory of ^ exhibiting the a-accessibility of ^.
Writing G for the left adjoint to F, each object G{B)^ B e ^, is as-
presentable for some regular cardinal a^. We fix a regular cardinal C
bigger than a and each a^. We shall prove that F has rank C.
Let A = colimit/ Ai be a /3-filtered colimit in j^. Since each B e^ and
the corresponding G{B) G si are /?-presentable (see 5.1.2), one deduces
from the adjunction G -\ F (see 3.1.5, volume 1):
^(B, F{co\imAi)) ^si(G{B),co\imAi)
^ cqlim j^{G{B),Ai)
^ coliai^(B,F{Ai))
^^(B,colimF(Ai)).
Since this isomorphism holds for every B G ^, one gets the isomorphism
F(GoUmt€7 Ai) = colimieAi F{Ai)\ see 5.3.8. D

5.5 Functors with rank 275
Putting together the previous results, we get the following form of the
adjoint functor theorem (see 3.3.3, volume 1).
Theorem 5.5.7 Let F: s/ >^ be a functor between two locally-
presentable categories. The following conditions are equivalent:
A) F has a left adjoint;
B) F has rank and preserves small limits.
Proof Locally presentable categories are accessible, complete and co-
complete (see 5.3.2 and 5.2.8).
If F has a left adjoint G, F preserves small Umits (see 3.2.2, volume 1)
and has rank (see 5.5.6). Conversely if F has rank it satisfies the solution
set condition (see 5.5.4) and thus has a left adjoint (see 3.3.3, volume 1).
D
Using the same type of arguments, we can now strengthen proposition
5.3.3.
Theorem 5.5.8 Let s/ be an accessible category. The following
conditions are equivalent:
A) si is locally presentable;
B) si is cocomplete;
C) si is complete.
Proof The equivalence between A) and B) is established by 5.3.3,
while the implication A) => C) is contained in 5.2.8. It remains to prove
C) ^ B).
Given a small category J>^ we consider the functor
A: si >Vm{J,si)
mapping an object A to the constant functor on A\ we must prove that
A has a left adjoint (see 3.2.3, volume 1). But Fun{J^^si) is complete
(see 2.15.2, volume 1) and since limits in Fun(y, j^) are computed point-
wise, A preserves limits. By 3.3.3, volume 1, it remains to prove that A
satisfies the solution set condition. This could be deduced from 5.5.5 via
the proof of the result that given a small category y, Fun(y, si) is still
accessible (see 5.7.5). The direct proof we give here is a crucial step in
the proof that fun{J,si) is accessible.
Thus we consider a small category y, a functor F: ^ > js/, an object
A^ si and a natural transformation 9: F =^ ^(^)- Each F(/), I e J,
is a/-presentable for some regular cardinal a/. Applying 5.4.5 and 5.4.8,
we raise the degree of accessibility of si to some regular cardinal a,
strictly bigger than all the a/ and the cardinality of the set of arrows of

276
Accessible categories
F{J)
Diagram 5.2
y. Using the notation of 5.3.1, we consider the full subcategory ^ C s/
exhibiting the a-accessibility of j/. We write A = coliia(^G,g) G, where
(G, g) G ^/A^ for the corresponding a-filtered colimit concerning A. We
recall that a-filtered colimits in Fun{J^^s/) are computed pointwise (see
2.15.1, volume 1).
For every object I E J^ the morphism 6i: F{I) >A factors through
some canonical morphism 5(g,p)- G >A^ since F{I) is a-presentable
(see 5.1.3). Since #y < a, by a-filteredness we can assume that the same
index (G, g) can be used for all objects I ^ J. Now given a morphism
i: I >J in */, we have diagram 5.2 where gogj =z Qj and gogj = 9j.
There is a priori no reason to have gj o F{i) = gi. But since
gogjo F{i) = 9j o F{i) = 9i =gogi^
5.1.3 implies the existence of some /: {G,g) >{G',g') such that the
relation f o gj o F{i) = f o gj holds. Since #y < a, by a-filteredness
we can again assume that the same {G\g') can be used for all arrows
i € t/. Putting 9'j = fogi^ we have thus defined a natural transformation
ff: F => A(G') such that A{g')o9' = 9. This proves that ^ is a solution
set condition for A at F. D
To conclude this section, let us come back to the case of monads with
rank, already studied in section 4.6.
Theorem 5.5.9 Let s/ be a locally presentable category and T =
{T,e,fx) a monad with rank on s/ (i.e. the functor T: s/ >s/ has
rank). Under these conditions, the category s/'^ ofT-algebras is locally
presentable as well.

5.6 Sketches 277
Proof We write U: s/'^ >s/ for the forgetful functor and F for its
left adjoint (see 4.1.4). The very difficult part of the proof is to show the
cocompleteness of s/^ but this has already been done in 4.3.6, since a
locally presentable category is both complete and cocomplete (see 5.2.8).
Choose now, by 1.3.3 and 1.5.2, volume 1, a regular cardinal a such
that s/ is locally a-presentable and T has rank a. By 5.2.8, U preserves
a-filtered colimits. Let us choose a family {Gi)i^i of strong generators
of s/^ each Gi being a-presentable.
The functor j^'^{F{Gi), -): j^'^ >Set is, via the adjunction F -\U,
isomorphic to the functor ^{Gi^ U—). But s/{Gi^ —) preserves a-filtered
colimits since Gi is a-presentable and U preserves a-filtered colimits by
5.2.8. Thus the composite si{Gi,U-) = j^'^{F{Gi),-) preserves a-
filtered colimits and F{Gi) is thus a-presentable in s/'^.
Moreover U reflects isomorphisms (see 4.1.4) as well as the family of
functors j2/(Gi,—), i e I^ since the G^'s constitute a family of strong
generators in j^] see 4.5.10, volume 1. Therefore the family of
functors s/{Gi,U—) = s/'^(^F{Gi),—), i e I, reflects isomorphisms and
(^F{Gi))^ J is thus a family of strong generators in s/'^. D
5.6 Sketches
In section 5.3, we have characterized the accessible categories as being
the categories of a-flat functors ^ >Set, for some small category
^. In this section, we shall investigate another characterization: the
accessible categories are also the categories of models for some "sketch".
Definition 5.6.1 A sketch is a triple S = {^,SP,J) where:
A) S' is a small category;
B) V is a set of cones on functors F: Qi >^, defined on small
categories 2;
C) I is a set of cocones on functors F: S) >^, defined on small
categories S).
Definition 5.6.2 Let S = (^,P,X) be a sketch. A model ofS is a
functor M: ^ >Set with the following properties:
A) if {pd: T >F{D))j^^^ is a cone of V on the functor F:
2 >^, then {M{pd):M{T) >MF(D))^^^ is the limit
ofMoF;

278 Accessible categories
B) if {sD: F{D) > T) ^^^ is a cocone of I on the functor F: Q) > ^,
then {M{sd): Mf[d) >M{T))j^^^ is the colimit ofMoF.
A morphism ofS-models is just a natural transformation. We write Mods
for the category ofS-models.
The definition of a sketch S = {^^V^I) could be generaUzed in a
rather straightforward way by replacing the category ^ by a graph ^,
together with a set C of commutativity conditions (see 5.1.1 and 5.1.5,
volume 1). But applying 5.1.6, volume 1, one observes immediately that
replacing the pair (^, C) by the induced category ^ does not change the
category of models.
On the other hand, definition 5.6.1 could have been strengthened by
requiring that all the cones in V be already limit cones and all the
cocones in I be colimit cones. This again does not change the range of
categories of models, as corollary 5.6.7 shows.
We shall prove that the category of models of a sketch is accessible.
For this, we need a lemma that people trained in logic will recognize as
a form of the downward Lowenheim-Skolem theorem.
Lemtna 5.6.3 Let S = {^,V,X) he a sketch. There exists a regular
cardinal a such that for every model M G Mods and every family of
elements {xq G M{Ti))^ j, with^f^I < a, there exists a submodel N C M
containing all the elements Xi and such that #(IJtg.^^-^) ^ ^*
Proof Let us choose first a regular cardinal f3 which is strictly larger
than all the following:
• the cardinality of the set of arrows of ^;
• the cardinality of the set V]
• the cardinality of the set X;
• the cardinality of the set of arrows of each small category S> which
appears in the definition of a cone of V;
• the cardinality of the set of arrows of each small category Si which
appears in the definition of a cocone of I.
We define a to be the regular cardinal a = (/3^^)"^; see section 5.4.
Clearly, this definition makes sense because of all the smallness
conditions in the definition of a sketch.
Now let M: ^ ^-Set be a model of S. A family {XT)Te^ of subsets
Xt Q M(T) constitutes the values of some submodel M' C M, i.e.
Xt = M'{T)y precisely when the following conditions are satisfied.

5.6 Sketches 279
A) Functoriality condition: for every arrow t: T' >T of ^ and
element X G Xt', M{t){x) G Xt.
B) Limit condition (see 2.8, volume 1, for the description of limits in
Set): for every cone (p^: T ^^iD))r}es> ^^ ^ ^^^ every
compatible family of elements (xd ^ ^fcd))^,^^? ^he unique element
X G M(r) such that M{pd){x) = xd for each D e 2 belongs
to Xt\ here compatibility means {MF){d){xD) = xd' for every
d: D >D' m9),
C) Colimit condition (see 2.8 and 2.4.6.b, volume 1, for the description
of colimits in Set): for every cocone [sd'- F{D) ^'^)tg^ ^^ ^''
(a) every element x G Xt has the form M{sD){y) for some D e S^
and y G X/r(D);
(b) if two elements x G Xf^d) and 2/ G A'/r(£)/) are such that
M{sd){x) = M{sD'){y)^ there exists a finite sequence D =
Dq^. .. ^Dn = D' oi objects of ^ and a corresponding sequence
of elements Xi G X/r(£).), xq = x, Xn = 2/ such that for each pair
(xi, Xi+i) of consecutive elements, there exists d: Di )• A+i in
S> such that {MF){d){xi) = x^+i or there exists d: A+i ^ A
in ^ such that (MF)(d)(xi+i) = x^.
For the sake of brevity, a sequence x = xq, ..., x^,..., Xn = y of elements
as in C)(b) will be referred to as a finite sequence of elements exhibiting
the equivalence of x and y.
It is quite straightforward to realize conditions A) to C). For this we
introduce various constructions, where M: ^ ^•Set is thus a model of
S and Xt C M{T) for each Te^.
• Construction f: define Xj^, for T G .^, by the formula
X(. = {M{t){x)\t: r ^Tin^, xGXt/}.
Clearly, Xt C XjI, just by choosing t = It-
• Construction I: for each cone (p^: T >F{D))^^^ in V and each
compatible family (x£) G Xp^^))^ ^, add to Xt the unique element
X G M{T) such that M(pd)(x) = xd for each J9 G ^; write X^ for
this bigger set containing Xt-
• Construction c^: for each cocone (^i^: F(£)) '^'^)dg3i ^^^ ^^^'^
element x G Xt, choose one index D e S> and one element y G
{MF){D) such that M(sjr>)B/) = x; add this element y to Xi?(x));
write X<f., for the bigger set containing Xt' obtained at T' G «^.

280 Accessible categories
• Construction c^: for each cocone [sd'- F{D) '^'^)dg3) ^^^ ^^^
pair of elements x G Xf{d)', V ^ ^F(r>0 such that M{sd){x) =
M{sD'){y)i choose one sequence x = xo,..., x^,..., x^ = y of
elements Xi G {MF){Di) exhibiting the equivalence between x and
y and add each Xi to each Xf^^.)] write X^., for the bigger set
containing Xt' obtained at V G ^.
The crucial point is now to verify that all the previous constructions
are compatible with the choice of the regular cardinal a; more precisely
where * stands for any of the four constructions. Let us thus assume
if^rp^^Xr < OL and prove the corresponding conclusion for each of the
four constructions. Observe that "< a" is equivalent to "< /3^^" by
definition of a.
Construction f: Given T e ^^ there are strictly less than a arrows
t: T' >T and strictly less than a elements in each Xt^; thus #X<j, < a
by regularity of a.
Construction I: For each cone (p^: T ^F{D))^^^ in V, if^S^ <
13 and #Xf(d) ^ Z?^^? for every D e S^. The number of compatible
famiUes is thus bounded by (/3^^)^^ = /3^^ (see 5.4), so is strictly less
than a. But #-X't < a and there are strictly less than C < a cones
to consider; by regularity of a, this implies #X,j. < a and thus also
Construction c^: For each cocone {sd'- F{D) ^'^)dg^ ^^ -^ ^^^ ^^^
ment (at most) is added to the disjoint union Utg.^^^' ^ ^'^^ number
of elements added is strictly less than /3 < /3^^,
E^e^#^T = *]lTe^^T <I3<^ + I3 = 0<^ < a.
Construction (?: For each cocone {sb'- F{D) '^'^)dg9 ^^ -^' there
are at most (P^^)^ = f3^^ pairs (x, ?/) to consider and for each pair of this
kind, finitely many elements are added to the disjoint union TTy^^y-^T?
i.e. strictly less than Hq elements. Thus for a given cocone, at most
(i9<^) • i^o) < {0^^){/3<^) = /3^^ elements axe added. So we start
with strictly less than a elements for each cocone in a family J with
#J < j9 < a; by regularity of a, we end up with strictly less than a
elements.
We are now ready to prove our statement. Let us choose a family

5.6 Sketches 281
{xi e M{Ti)).^j in an S-model M: ^ >Set, with #X < a. We define
XT = {xi\ieI, XieM{T)}
for every T G ^. By assumption, 5^x^^#^t ^ #2^ < ce- We construct
now a transfinite sequence of families (X/^ C M(r))^ ^ :
A) X^ = Xt;
B) if {Xj^)rp ^ is defined, (X^"^^)^ ^ is obtained by successively
applying the four constructions fj^c^^c^ to (X^)^ ^;
C) if A is a limit ordinal, we define Xj, = [j^^^^Xj. for every T e ^.
We shall prove that the required submodel A^ C M is given by N{T) =
Xj, , for every T G ^; as usual, we identify a cardinal with the smallest
ordinal of this cardinality, which is thus a limit ordinal.
First of all, let us prove by transfinite induction that 5^x^^#iV(r) <
a. This is true at the level 0 of the construction and we have already
observed that the four constructions /,/,c^,c^ respect boundedness by
a. Now if A < C^^ is a limit ordinal, then A < a and assuming that
Ylre^'^'^T ^ ^ f^^ every z>' < A, we get Ylre^'^-^T ^Y regularity of a.
By construction, N contains the original family {xi)i^i and it remains
to check that A^ is a submodel of M. By the description we have made
of a submodel, see conditions A), B), C)(a), C)(b), this is equivalent to
proving the stability of N under each of the four constructions fj^c^^c^.
Construction f: If t: T' >T is an arrow of ^ and x G N(T'), then
X G X^, for some A < C<^ and thus M(t){x) G X^+^ C N{T).
Construction I: Given a cone {po'- T ^^(^))dg^ ^^ ^ ^^^ ^
compatible family of elements (^xd ^ ^^^)^^))dg3i'> ^^^ element xd
belongs to some X^^j^y for \d < C^^- But since #^ < C
see section 5.4. This implies that we can choose A < C^^ with Ad < A for
all J9 G ^. But the unique element x G M{T) such that M{pd){x) = xd
lies in X^+^ C N(T).
Construction c^: Given a cocone {sd'- F{D) ^^)dg^ ^^"^^ ^^ ^^~
ement x e N{T) belongs to some X^, for A < C^^. Therefore we get
De^andye X^+^) C {NF){D) such that M{sD){y) = x.
Construction <?: Given a cocone {sd\ F{D) ^^)d€^ ^^^ *^^ ^^^
ments x e {NF){D), y G {NF){D') such that M{sd){x) = M{sD'){y),

282 Accessible categories
we have x G X^(^) and y G X^^^'^,), for A' < /3<^ and A'' < /3<^.
Writing A for the bigger of the two ordinals A', A'', we have x G Xp^^^^ and
y ^ -^FiD')' Therefore in (X^"^^)^ ^, and thus in N^ there is a finite
sequence exhibiting the equivalence of x and y. D
Proposition 5.6.4 The category of models of a sketch is accessible.
Proof Let S = (^, 'P, X) be a sketch. We consider the regular cardinal a
constructed in lemma 5.6.3 and prove that the category Mods of models
is a-accessible.
The category Mods is a full subcategory of Fun(^, Set); let us prove
first that Mods is closed in Fun(^,Set) under a-filtered colimits.
Consider for this a a-filtered small category ^ and a functor F:
3F )>Mods; let us write [sx'- F{X) ^^)xe^ ^^^ ^^^ colimit of F
in Fun(^, Set), which is computed pointwise by 2.15.2, volume 1.
Choose a cone {po: T >G{D))j^^^ on G: Q) ^^, in the set V
of distinguished cones. For every object X G ^, the composite
Q) ^^—^Sr ^^^^ >Set
admits
(f{X)(p^): Fix)[T) >F{X){G{D)))^^^
as a limit, just because F{X) is a model of S. But ^ is an a-category and
^ is a-filtered; therefore by 6.4.5, volume 1, one gets the isomorphism
cdim(limz)e^F(X) (G(Z?))) ^ limz)e^(colimF(X) (G(Z?))).
But since the colimit L of the functor F in Fun(^, Set) is computed
pointwise,
colimF{X){G(D))^L{G{D)).
On the other hand since F{X) is a model of S,
\imDe^F{X){G{D)) ^ F{X){T).
Thus the previous isomorphism yields
L(r)-conmF(x)(r)
^colim(]imne^F{X){GiD)))
^limo€^(coUmF(X)(G(D)))
^ limDg^L(G(£>))

5.6 Sketches 283
which is the required condition. The proof concerning a cocone of I is
completely analogous, applying 2.12.1, volume 1, instead of 6.4.5,
volume 1.
The previous argument shows that L is indeed a model of S, proving
that Mods has a-filtered colimits computed pointwise.
We must now define a set {Mi: ^ ^Set)^^/ of S-models exhibiting
the a-accessibility of Mods as in lemma 5.3.2. We choose those models
M such that the set IJtg.^-^(-^) '^^ strictly less than ot elements. The
elements of this set are in bijection with the objects of the category
Elts(M) (see 1.6.4, volume 1) and since ^ itself has strictly less than a
arrows, so has Elts(M), by the regularity of a. By 2.15.6, volume 1, we
know that each M is the colimit of the functor
Elts(M) ^—>2r ^—>Fun(^,Set)
where 0 is the forgetful functor and Y the Yoneda embedding. Since
there is - up to isomorphisms - just a set of categories with strictly
less than a arrows, this implies that - up to isomorphisms - there is
just a set of such functors M with #IJtg.^-^(-^) ^ ^' more precisely,
choose one functor M in each isomorphism class. We write ^ for the full
subcategory of Mods generated by those models M.
By the Yoneda lemma, the represent able functor ^(T, —) represents
on Fun(^, Set) the evaluation functor at T. Since colimits are computed
pointwise in Fun(^,Set) (see 2.15.2, volume 1), this functor represented
by ^(r, —) preserves all colimits. In particular each functor ^(T, —) is
a-present able in Fun(^,Set) and thus so is every functor obtained as
a a-colimit of representable functors (see 5.1.4). By 2.15.6, volume 1,
again, this shows that each of our models M G ^ is a-presentable in
Fun(^,Set). But a-filtered colimits are computed in the same way in
Mods and in Fun(^, Set), since /3 < a] thus each M G ^ is a-presentable
in Mods.
Finally we shall prove that each model M G Mods is the a-filtered
union (and thus colimit) of its submodels M' C. M which are (up to
an isomorphism) in ^. This will yield the conclusion by 5.3.2. First of
all, by 5.6.3, every element x G M(r), T e ^, belongs to a submodel
M' G ^, which proves already that the union is M itself. Moreover,
given a family (M^ C M)keK of submodels, with #jFC < a, the union
^' = UkeK^k in Fun(.^,Set) is still such that #LItg.^^'(^) < ^'
by regularity of a. Thus by 5.6.3 again, M' is contained in a submodel
M'' G ^. This proves the a-filteredness of the union. D

284 Accessible categories
EltSM "^^ > ^* -^UFun(^,Set)
Ji{-,M)
Set
Diagram 5.3
The rest of this section is now devoted to proving the converse of
5.6.4, namely that every accessible category is the category of models
of some sketch. The core of the proof is the following lemma which is a
classifying topos-like theorem (see 4.1.7, volume 3).
Lemma 5.6.5 Let Jt be an a-accessible category; write ^ for the full
subcategory of a-presentable objects. Then M is equivalent to the
category of those functors Fun(^, Set) >Set which preserve a-limits and
small colimits.
Proof By 5.3.8, ^ is equivalent to a small category, so that there is no
restriction in supposing in the proof that ^ is a small full subcategory
of JK containing exactly one object in each isomorphism class of a-
presentable objects.
For each object M G ./#, we can consider the Kan extension Km of
the functor
^(M, -): ^* >Set, G ^ Ji{G, M)
along the Yoneda embedding, as in diagram 5.3. Let us write EIIsm
for the category of elements of ./#(—, M) and 0m-* EIIsm >^ for the
corresponding forgetful functor. Applying 3.8.1, volume 1, and the
formula for computing Kan extensions pointwise (see 3.7.2, volume 1), we
get Km{F) = colimF o 0^, for every functor F: ^ >Set. The
interchange property for colimits (see 2.12.1, volume 1) and the previous
formula imply immediately that Km preserves all small colimits. On the
other hand ./#(—, M) is a-flat (see proof of 5.3.5), thus Km preserves
a-limits (see 6.4.13, volume 1).
Let us also recall that Km oY = ^{—^M) since Y is full and faithful
(see 3.7.3, volume 1). Given two objects M^M' G Ji^ the definition of
a Kan extension (see 3.7.1, volume 1) and the proof of 5.3.5 yield the

5.6 Sketches
Elts(F) "^^ > ^* —^Fun(^,Set)
GoY
Set
Diagram 5.4
285
isomorphisms
This proves already that J^ is equivalent to the category of functors Km
and natural transformations between them.
It remains to prove that each functor G: Fun(^,Set) >Set which
preserves a-limits and small colimits is isomorphic to some Km- Let us
consider diagram 5.4 where F is some functor F: ^ >Set, Elts(F) is
its category of elements and 0f the corresponding forgetful functor. We
know that F = coliTaYo(j)p (see 2.15.6, volume 1), while the Kan
extension LanyG of G along Y is given by the formula (see 3.7.2, volume 1)
(LanyG)(F) ^ colimG o Y o (pF = G(colimy o (^f)
= G{F),
since G preserves small colimits. Thus LanyG is just G, which preserves
a-limits by assumption. Therefore Goy is a-flat (see 6.4.13, volume 1)
and thus isomorphic to some Ji{—^M) by 5.3.5. So G is isomorphic to
Km. □
Lemma 5.6.5 can be interpreted as the fact that the accessible
category ^ is equivalent to the category of models of a "large sketch"
S = (^,P,X), where ^ is the (large) category Fun(^,Set), V is the
class of all a-cones and I is the class of all small cocones. It remains
to reduce sufficiently the size of those classes to end up eventually with
just sets.
Theorem 5.6.6 A category is accessible if and only if it is the category
of models of some sketch.

286 Accessible categories
Proof One implication has been proved in 5.6.4. To prove the converse,
we start from 5.6.5, from which we borrow the notation. In particular
J( is a-accessible and ^ is small and equivalent to the subcategory of
a-presentable objects.
Up to isomorphisms, there is just a set of categories with strictly less
than a arrows, and thus just a set of functors H: S) >^* defined on
such categories 2). For every functor of this kind, we choose one
"distinguished" limit cone [pd- L >YH{D))j^^^ on F o if in Fun(^,Set),
3) ^—>T ^—>Fun(^,Set),
where Y is the Yoneda embedding.
Our category ^ is the full subcategory of Fun(^,Set) whose objects
are the functors ^(G, —), for G G ^, and the objects L, chosen as
distinguished limits of the a-diagrams of representable functors. Clearly, ^ is
small.
As set ^ of cones, we choose for each a-diagram of representable
functors the corresponding distinguished limit cone {po'- L >YH{D))^ ^.
As set of cocones, we choose, for each object F G ^, the canonical
eocene ^/F. In other words we consider the category Elts(F) of elements
of F and put in X the canonical colimit cone of
Elts(F) —^^—^ T ^—> Fun(^, Set)
with vertex F (see 2.15.6, volume 1).
By construction of the sketch S = (^,7^,X), we have:
A) ^ is a full subcategory of Fun(^,Set);
B) every representable functor belongs to 2r\
C) every cone of P is a a-limit cone in Fun(^, Set);
D) every cocone of X is a small colimit cone in Fun(^,Set);
E) for every a-category Q) and every functor H\ Q) >^*, V contains
a limit cone on the functor y o if;
F) for every functor F G ^, X contains a colimit cone on the functor
Yo<\>p,
We shall prove that M is equivalent to the category of models of the
sketch § = (.^,P,X).
By the first condition in the preceding list, we can consider the full
embedding i\ ^ ^--^ Fun(^, Set). By the second condition, the Yoneda
embedding factors as y = z o y', with Y'\ ^* >^, Conditions C)
and D) imply that given M ^ Jl, Km o i'- ^ >Set is a model of

5.6 Sketches 287
S, just because Km preserves a-limits and small colimits. In the same
way a natural transformation 9: Km =^ Km' yields another natural
transformation 9 *i: Km oi =^ Km' o i- Combining this with 5.6.5, we
have already defined a functor (p: Ji >Mods.
Every functor F\ ^ >Set can be written as F = colimF o c^p^
i.e. as a colimit of representable functors (see 2.15.6, volume 1). Given
M, M' G M^ Km and Km' preserve this colimit so that a natural
transformation 9\ Km => Km' is completely determined by its values on the
representable functors. Since ST contains the representable functors, 9 is
thus determined by & * i, proving the faithfulness of (^.
To prove that ^ is full, we must check that every natural
transformation r: Km ^i => Km' o i has the form 0 * i for some 9\ Km => Km'-
Again write F: ^ >Set as F = colimF o 0;r. By naturality of r, we
get a cocone
KMoYo(l)F{A,a) ^^'^ >KM'oYo(l>F{A,a) ^^'^""^KM'iF)
where (A, a) runs through Elts(F) and a corresponds with a by the
Yoneda lemma. This yields a unique factorization 9f^
Km{F) ^ XM(colimy o (Pf) ^ colim^M o y o c^f >Km'{F),
such that 9f o -K^m(^) = KM'ifi) o ty'a for every (A, a) G Elts(F). By
naturality oi r^ 9f = tf when F e ^. On the other hand 9 is natural
since given a: F =^ G in Fun(^, Set), the relation 9g^Km{cf) = Km'(sf)^
9f follows from the Yoneda lemma via the equalities
9g o Km{(^) o Km {a) = 9go Km (crA(a))
= KM'{crA{cL)) OTy^A
= Km' (ct) o Km' {q) oty'a
= Km'{(^) ^ 9f ^ Km^p)
when (A, a) runs through Elts(F); indeed the morphisms a constitute a
colimit cone (see 2.15.6, volume 1) preserved by Km- Thus 9 is natural
and r = 0 * i
To prove that (^ is an equivalence, it remains to show that every model
R: ^ >Set of S is isomorphic to the restriction of some Km- Given an
S-model R: ^ >Set, we extend it to a functor R: Fun(^, Set) >Set
in the following way. A functor F: ^ >Set can be written as F =
colimy o (l)F] we define 'R{F) ^ colimii o Y' o 0;r. By 3.7.2, volume 1,
R{F) ^ Lany(-R o Y'){F) and thus by 3.8.1, volume 1, one also has
R{F) = colimF o (J)roY'' The interchange property for colimits (see

^ Accessible categories
2.12.1, volume 1) implies immediately that R preserves colimits. Now if
F e ^^ the colimit cone of F o 0ir is in I so that
R{F) ^ colimi? o Y'o())F = i?(colimy' o cPp) ^ R{F).
This proves that ^ extends R.
Let us prove now that R preserves a-limits. Since R is the Kan
extension oi RoY^ along F, this is equivalent to proving that this
composite RoY^ is a-flat. The proof is analogous to the implication A)
=> C) in 6.4.13 (i.e. in 6.3.6) volume 1. Since we work on ^*, we
must prove the a-filteredness of Elts(i? o F'). Given an a-category ^
and a functor H: Q) >^\\s[RoY'), write HD = (A^^ar)). The limit
{Pd- L >5^^d)dg^ of y o (j)RoY' o if is a cone in V^ by assumption.
Thus
YimDe^RYiAo) ^ R{YimDe9Y{AD)) ^'R{YimDe9Y{AD))
= colim(limDe^5^(^r>)) o (t^RoV-
Since {aD)De^ is a compatible family in \iin.De^^Y{Ao)^ it is
represented by some element in some term of the colimit. This means the
existence of an object (A, a) e Elts(i2 o y') and a compatible family
of morphisms [fo G ^{Ad^A))^^. In other words, we have found a
cocone (/d: (AD^ao) >{A,a))^^^ on H.
Finally R preserves all small colimits and all a-limits, proving by 5.6.5
that it is isomorphic to some Km- And since the restriction of i? to ^
is just R, this ends the proof that (f is an equivalence of categories. D
Corollary 5.6.7 Every sketch has a category of models equivalent to
the category of models of a sketch S = (^,P,X), where V is a set of
hmit cones and X is a set of colimit cones.
Proof The category of models of a sketch is accessible (see 5.6.4) and
an a-accessible category can be presented as the category of models for
a sketch S = (,^, V, I) where P is a family of a-limit cones and I a
family of small colimits (see proof of 5.6.6). D
Corollary 5.6.8 Consider a sketch S = (^,P,X). IfV or I are empty
the category of models of S is locally presentable.
Proof We treat the case V empty; the other is analogous, using colimits
instead of limits. By 5.6.6 and 5.5.8, it suffices to observe that Mods is
stable in Fun(^, Set) under small limits, which follows immediately from
2.12.1, volume 1. D

5.7 Exercises 289
Theorem 5.6.6, disregarding the fact that it is written in terms of
equivalent conditions, does not give any information on the degree of
accessibility. For example, consider a small category ^ with finite limits
and colimits. The category of those functors F: 3' >Set preserving
finite limits and colimits is accessible, as attested by 5.6.4. But what is
its degree of accessibility? The cardinal ot constructed in 5.6.3 is so big
that it does not give much precise information. Another example: start
with a /^-accessible category; as in 5.6.6, present it as the category of
models of a sketch; then using this sketch, construct the cardinal ot as in
5.6.3; OL is in general much bigger than «:, while the accessible category
which is considered is still the same.
5.7 Exercises
5.7.1 Let ./# be a category such that both Ji and its dual M'^ are locally
presentable; show that Jt is equivalent to a poset.
5.7.2 Show that the category of Hilbert spaces and linear contractions
is ^<o-accessible and equivalent to its dual (compare with 5.7.1).
5.7.3 Show that the category of sets and partial bijections is ^<o-ac-
cessible and equivalent to its dual (compare with 5.7.1). By a partial
bijection from A to B we mean a relation R from A\^o B which is the
graph of a bijection /': A! >B', with A! (Z A and B' C B.
5.7.4 Show that every Cauchy complete small category is accessible.
[Hint: take a infinite and larger than the cardinality of the category and
prove a"^"^-accessibility.]
5.7.5 Let y be a small category and Ji an accessible category. Show
that the category Fun{^^Ji) of functors and natural transformations is
accessible.
5.7.6 Let Ji be an accessible category and a a regular cardinal. Show
that the category of a-presentable objects of Ji is equivalent to a small
category.
5.7.7 Show that an accessible category with pushouts is co-well-powered.
5.7.8 Prove that the relation a</3 between regular cardinals is transitive.
5.7.9 Consider regular cardinals a</3 and an a-accessible category Ji.
Prove that an object M G .^ is ^-presentable iff it is a a-filtered colimit
of a-presentable objects over a diagram of size strictly less than C.
5.7.10 Given regular cardinals a,/3, prove that one has a < ^ iff every
a-accessible category is ^S-accessible.

290 Accessible categories
5.7.11 Let J2/, ^ be accessible categories with s/ small. Prove that if ^
is complete (resp. cocomplete), the category of functors with rank from
j/ to ^ is complete (resp. cocomplete).
5.7.12 Prove that the category of small accessible categories and
functors with rank is complete.
5.7.13 If ^2/,^,^ are accessible categories and F: si >^, G: ^ >^
are functors with rank, prove that the comma category (F, G) is
accessible.
5.7.14 Let Si = (^i,Pi,Ji) and S2 = (^2,7^2,^2) be sketches. A
functor F\ S'x >^2 is a morphism of sketches when it takes each
cone of V\ to a cone of V2 and each cocone of X\ to a cocone of X2- For
such a morphism of sketches, prove that composition with F yields a
functor with rank Modsi >Mods2-
5.7.15 If S = (ST^ 'P,X) is a sketch, a model of S in some category ^ is
a functor F\ 3^ >^ mapping the cones of V to limit cones and the
cocones of X on colimit cocones. Prove that the models of S in a locally
presentable category constitute an accessible category.

6
Enriched category theory
A category si consists of:
• a class of objects \si\\
• for every pair A, B of objects, a set sii^A^ B) of arrows;
• for every triple A, B, C of objects, a composition law
s/{A,B) X j/(B,C) >si{A,C),
these data satisfying the identity and associativity axioms.
In 7.1.1, volume 1, we have seen that in the case of a 2-category,
the sets s/{A^B) of arrows turn out to be provided with an additional
structure: that of a small category, while the composition respects this
structure in each variable. In 1.2.1, an analogous situation has been
encountered for preadditive categories: the sets s/{A^B) are now provided
with the structure of an abelian group while the composition respects
this structure in each variable. We shall say that a 2-category is
"enriched over the category of small categories" while a preadditive category
is "enriched over the category of abelian groups".
Preserving a structure in each variable is something which can be
easily expressed for a mapping X x Y >Z, as long as it makes sense to
fix an element of X or y. This contradicts the very spirit of category
theory, where objects are basic entities. Now observe that when X, y, Z
are small categories, being functorial in each variable is just being
globally functorial (see 1.6, volume 1). And when X, Y, Z are abeUan groups,
the biadditive mappings X x Y >Z on the product are in bijection
with the group homomorphisms X <S> Y >Z on the tensor product.
In this chapter we shall introduce first the so-called "symmetric mon-
oidal closed categories", which are the best substitutes for Set for
developing enriched category theory. Then we shall indicate how to enrich
291

292 Enriched category theory
a ABC ® 1
{A(S){B® C)) (8) D
(^A,B,C®D
A®i(B®C)® D),-— >A ®iB®(C® D))
^CLBCD
Diagram 6.1
over them the basic concepts and results of category theory.
It is a deliberate choice, in this chapter, to reduce the proofs to the
necessary constructions and the key arguments. We hope this will clarify
the spirit of enriched category theory, even if the conscientious reader is
left with a large amount of routine calculations.
6.1 Symmetric monoidal closed categories
Definition 6.1.1 A monoidal category 'V consists in giving:
A) a category i^;
B) a bifunctor ®: i^ x i^ >i^, called the tensor product - we write
A® B for the image under ® of the pair (A, B);
C) an object I ei^, called the unit;
D) for every triple A^B^C of objects, an ''associativity^^ isomorphism
dABC' {A®B)® C >A ®{B® C);
E) for every object A, a ''left uniV^ isomorphism
I A'- I ® A >A\
F) for every object A, a "right unit" isomorphism
r^: A ® I >A.
These data must satisfy the following requirements:
A) the morphisms aABC s^re natural in A, B, C;
B) the morphisms I a are natural in A;
C) the morphisms va are natural in A;
D) diagram 6,1 is commutative for every quadruple of objects A, B, C,
D (associativity coherence);

6.1 Symmetric monoidal closed categories
{A®I)®B ^^^^ yA®{I®B)
293
r^ ® 1
1®Ib
A®B
Diagram 6.2
{A®B)®C —^^B®1 ^ {B®A)®C
dABC
A®{B®C)
SA,B<S>c\
{B®C)®A
dBCA
Diagram 6.3
dBAC
B®{A®C)
1 ® SAC
^ B®{C®A)
E) diagram 6.2 is commutative for every pair A^B of objects (unit
coherence)
Definition 6.1.2 With the notation of 6.1.1, a monoidal category is
symmetric when, moreover, an isomorphism
SAB' A ® B >B ® A
is given for every pair A, B of objects. These isomorphisms must be such
that:
A) the morphisms sab are natural in A, B;
B) diagram 6.3 is commutative for every triple A^B^C of objects
(associativity coherence);
C) diagram 6.4 is commutative for every object A (unit coherence);
D) diagram 6.5 is commutative (symmetry axiom) for every pair A^B
of objects.
Definition 6.1.3 With the notation of 6.1.1, a monoidal category i^ is
biclosed when, for each object B ei^, both functors
-®B:'r >Tr, B®-:i^ >i^

294
Enriched category theory
Diagram 6.4
sba
A<S>B
Diagram 6.5
have a right adjoint. A biclosed symmetric monoidal category is called
a symmetric monoidal closed category
Since in a symmetric monoidal category, both functors — 0 B and
B 0 — are naturally isomorphic, one has obviously
Proposition 6.1.4 A symmetric monoidal category i^ is closed if and
only if, for each object B ei^, the functor — <S> B: "V >Y has a right
adjoint. D
Let us immediately introduce a crucial example.
Definition 6.1.5 A category Y is cartesian closed when it admits all
finite products and, for every object B e i^, the functor -xB: i^ >i^
has a right adjoint, generally written (-)^: i^ >i^.
Proposition 6.1.6 Every cartesian closed category is symmetric
monoidal closed, with the cartesian product as a tensor product.
Proof The existence of the required isomorphisms and the various
coherence conditions follow immediately from the universal property of
products (as in 2.1.6, volume 1). The unit is just the terminal object.
D
In the rest of this chapter, we shall most often reduce our attention
to the case of symmetric monoidal closed categories. As long as this can
be done at no extra cost, we shall nevertheless give definitions or state
results in a more general context.

6.1 Symmetric monoidal closed categories 295
Let us now introduce some more notation. Given a monoidal category
i^, the functor i^{I, -): i^ >Set represented by the unit / is generally
called the "forgetful functor" or "underlying set functor". Sometimes we
just write it U. We make a strong point of the fact that this forgetful
functor is in general not faithful: for example, when i^ is the cartesian
closed category of small categories, / is the one point category and thus
i^{I, A) is - up to isomorphism - just the set of objects of the small
category A. And clearly a functor is not determined by its action on the
objects.
When i^ is a. symmetric monoidal closed category, we write
[B,-]:^ >^, C^[B,C]
for the right adjoint to the functor — 0 B: i^ >i^. In particular the
isomorphisms
r{B, B) ^ r{i ®B,B)^ r{i, [b, b])
yield a "unit" morphism ub'- I >[B^B] corresponding with the
identity on B. In an analogous way the isomorphisms
r{c, c) ^ r{c ®i,c)^ r{c, [/, c])
yield a morphism ic'- C ^[^jC'] corresponding with the identity on
C It is also useful to consider the "evaluation morphisms"
evAB*. [A,B]®A >B
corresponding by adjunction with the identity on [A, B] and the
"composition morphisms"
CABC- [A, B] ® [B, C] ^ [A, C]
corresponding by adjunction with the composite of diagram 6.6.
Proposition 6.1.7 On a symmetric monoidal closed category i^, we
get a bifunctor
[-,-]: r*xr >r, {a,b)^[a,b]
whose composite with the forgetful functor i^{I, —): i^ ^•Set is just
ir*xr >Set, {A,B)^r{A,B).
Proof Given /: A >A', [f,B]: [A\B] >[A,B] corresponds by
adjunction with the composite
[A\B] <S> A ^^^ y[A\B] (8) A^ ^^^'^ >B,

296
Enriched category theory
[A, B] ® [B, C]®A
[A,B]®A®[B,C]
\evAB ® 1
B®[B,C]
[B,C]®B
\evBC
c
Diagram 6.6
It is routine to check the functoriality.
On the other hand the isomorphisms
r{i, [B, c]) ^ r{i ®B,c)^ r{B, c)
prove the second assertion.
D
Proposition 6.1.8 In a symmetric monoidal closed category i^:
A) the morphisms ub'- I >[B, B] are natural in B;
B) the morphisms ic'. C > [/, C] are isomorphisms;
C) the morphisms ic: C > [/, C] are natural in C;
D) diagram 6.7 and diagram 6.8 are commutative, for all objects A, B,
C, D in tT.
Proof The inverse of the morphism ic is the composite
[/, c] -^iiu [/, c] ® I ^^^^ )a
The rest of the proof is routine computations left to the reader. D
Examples 6.1.9
6.1.9.a The category Set of sets and mappings is cartesian closed, as
observed in 3.1.6.f, volmne 1.
6.1.9.b The category Cat of small categories and functors is cartesian
closed, as proved in 3.1.6.g, volume 1.

6,1 Symmetric monoidal closed categories
CABC ® 1
{[A, B] ® [B, C]) ® [C, D] >[A, C] ® [C, D]
tt[A,B],[B,C],[C,D]
297
[A,B]®([B,C]®[C,D])
1 ® CBcr>
CAcr>
[^, B] ® [B, D] ^^^ > [A, D]
Diagram 6.7
[A,B]0I I0[A,B]
l^UB
[A,B]0[B,B]-
t^A ® 1
Diagram 6.8
6.1.9.C The category of models of a commutative algebraic theory
is symmetric monoidal closed (see 3.10.3); in particular the following
are examples of symmetric monoidal closed categories: abelian groups,
modules over a commutative ring, pointed sets, G-sets for a commutative
group G, ....
6.1.9.d If ^ is a small category, the category Fun(^, Set) of functors
and natural transformations is cartesian closed. Indeed, given two
functors G,H: ^ZZtSet, define
H^: ^-
^Set, CH^Nat(^(-,C)xG,if),
where this definition extends in an obvious way to the case of morphisms.
Given a third functor F: % >Set, write it as a colimit of representable
functors (see 2.15.6, volume 1). The conclusion follows now from the
isomorphisms
Nat(F xG,H)^ Nat((colim<<f(Ci, -)) x G,H^
^ Nat(coUm(«'(Ci,~) x G),h)

298 Enriched category theory
^limNat(^(Ci,~) xG,H)
^limNat(^(Ci,-),//^)
^Nat(colim^(Ci,-),//^)
^Nat(F,//^),
where we have twice used the Yoneda lemma and twice the fact that the
functor Nat(—,X) transforms colimits into limits (see 2.9.5, volume 1);
moreover for every object C e ^, the functor - x G{C): Set >Set
preserves colimits since it has a right adjoint (—)^(^) (see 6.1.7.a, this
volume and 3.2.2, volume 1); therefore — x G preserves colimits as well
since both products and colimits in Fun(^, Set) are computed pointwise
(see 2.15.2, volume 1).
6.1.9.e The case of G-sets, for G a group, is worth a comment. This
is a cartesian closed category, as a special case of 6.1.9.d (see 2.15.7.a,
volume 1). Thus we can compute the exponentiation of two G-sets B, C;
it is given by
C^ ^ G-Set(G xB,C)^ Set(B, C).
The first isomorphism comes from 6.1.9.d, this volume, and 2.15.7.a, vol-
lune 1, while the second one is easily proved. Indeed, given a morphism of
G-sets /: GxB >C, just consider the mapping / = /(I, -): B >C,
where the group has been written additively. This mapping completely
determines / since, given x e G, b e B,
f{x,b) = f{x{l,x-^b)) = xf{l,x^^b) = x7{x-^b).
Conversely every mapping g: B >C is induced in such a way by a
morphism g: G x B >G of G-sets: just put g{x, b) = xg{x~^b), which
yields of course g{l,b) = g{b). This p is a morphism of G-sets since,
given 2/ E G,
g{yx,yb) = yxg{x~^y~^yb) = yxg{x~^b) = yg{x,b).
6.1.9.f When the group G is commutative, the category of G-sets is
symmetric monoidal closed (see 3.10.3) for a structure determined by
[A,B]^G-Set{A,B)
for G-sets A, J5. But the category is also cartesian closed, for a structure
determined by
S^^Set(A,B);

6.1 Symmetric monoidal closed categories 299
see 6.1.7.e. This is an example of a category bearing two distinct
symmetric monoidal closed structures.
6.1.9.g The category of topological spaces and continuous mappings
can be provided with the structure of a symmetric monoidal closed
category: define [Y, Z] as the set of continuous mappings from Y to Z,
provided with the pointwise topology; the basic open subsets are
{y.U) = {f:Y >Z|/continuous, f{y)eU}
for 2/ G y and U open in Z. The corresponding tensor product X ®Y
of two spaces is the cartesian product of the two sets provided with the
final topology for the mappings
X >XxY, xi-^(x,2/),
Y >X xY y^ (x,2/),
for all elements x £ X^ y eY.
6.1.9.h The category Barii of Banach spaces and linear contractions
is symmetric monoidal closed for a structure characterized by
[B,C]^Banoc(B,C)
for two Banach spaces B^C (see 1.2.5.f, volume 1). It is indeed well-
known that Banoo(B, C) is a Banach space when provided with the norm
||/||=sup{||/F)|||6e5, ||6||<1}.
The tensor product A^S^ B oi two Banach spaces is the so-called
"projective tensor product" of A, B.
6.1.9.i If ^ is a small category, the category Fun(^, ^) of endofunctors
(i.e. functors from the category to itself) and natural transformations is
monoidal (but not symmetric, in general) when choosing the composition
of two functors as their tensor product.
6.1.9.J Every category with finite products is monoidal symmetric,
when choosing the cartesian product as tensor product. This structure
is closed just when the category is cartesian closed.
6.1.9.k If i? is a ring, the category of i?-i?-bimodules and left-right-i?-
linear mappings is monoidal biclosed, for the structure given by L<S>M =
L(®rM. The right adjoint to - (g) M is given by [M, —\r where
[M,N]r = {/: M >N\ f right-ii-Unear },

300 Enriched category theory
while the right adjoint to L 0 - is given by [L, -]i where
[L,N]i = {/: L >N\ f left-i2-linear }.
When R is not commutative, this is an asymmetric example.
6.1.9.1 A A-semi-lattice (A, <) with top element 1, viewed as a
category, admits the element a A 6 as product of the elements a,b £ A (see
2.1.7.b, volume 1). This category is cartesian closed as long as, for every
pair 6, c of elements, there exists an element b=> c such that
aAb<c iff a <b=> c.
In particular, every Heyting algebra (see 1.2.1, volume 3) is a cartesian
closed category.
6.2 Enriched categories
When a symmetric monoidal closed category is fixed, we show how to
enrich over it the notions of category, functor, natural transformation
and distributor. We use freely the notation of section 6.1.
Definition 6.2.1 Let i^ be a monoidal category. A i^-category^
consists in the following data:
A) a class \%\ of ''objects'';
B) for every pair A,B e \^\ of objects, an object ^{A,B) ofi^;
C) for every triple A^B^C £ \^\ of objects, a ''composition'' morphism
inr,
CABC- '€^A,B)®^{B,C) >^(A,C);
{A) for every object A e\^\, a "unit" morphism in i^,
ua:I >^(A,A).
These data must satisfy the following conditions:
A) given objects A, B, C^D^E e %, diagram 6.9 commutes
(associativity axiom);
B) given objects A^B e^, diagram 6.10 commutes (unit axiom).
When \^\ is a set, the i^-category % is called a small i^-category
As a first obvious result we get
Proposition 6.2.2 Let i^ be a symmetric monoidal category. With the
notation of 6.2.1, the i^-category % gives rise to a "dual" i^-category
** defined by:
A) 1*1 = 1*1;
B) <e''{A, B) = *(B, A) for A, Be *;

6.2 Enriched categories
301
(<g'(A, B) ® <i^{B, C)) ® <^(C, uy^^^^^^'^iA, C) O <^(C, D)
0^{A,B),'^(B,C),'g(C,D)
^{A, B) ® (-^(B, C) ® -^(C, Z)))
1 ® CBCD
'g'(A,5)®'g'(B,£»)
cacd
cabd
Diagram 6.9
->'g'(A,£))
I ® -^(A, g)'^^"^-^WcA, BK^^'^(A, B)®i
ua ® 1
1
<ig(A,B)
1®UB
'g'CA, A) ® <g'(A, B)-^^^^^(A, B)^^j^<giA, B) ® ns, B)
Diagram 6.10
C) ^\bc = ^CBA o s<^(B,A),^(c,B) for A,B,C £ ^*;
D) u*^ = UAforAe'^\
D
Definition 6.2.3 Let i^ be a monoidal category Given i^-categories
J2/,^, a y-functor F: si >0i consists in giving:
A) for every object Ae s/, an object F{A) G ^;
B) for every pair A, A^ £ s/ of objects, a morphism in "V,
Faa'- ^{A,A') >^{F{A),FiA'));
in such a way that the following axioms hold:
A) for all objects A^A'.,A" G si, diagram 6.11 commutes (composition
axiom);
B) for every object A e si, diagram 6.12 commutes (unit axiom).
Definition 6.2.4 Let i^ be a monoidal category. Let si^^ be two
i^-categories and F,G: si \^ two i^-functors. A i^-natural
transformation a: F => G consists in giving, for every object A E si, a
morphism
aA:I >^{F{A),G{A))
in y such that diagram 6.13 commutes, for all objects A^A^ e si

302 Enriched category theory
j^{A, A^) ® j^{A', A'') ^^^^^^ > j^{A, A'O
Faa' ® Fa'A"
Fa,a"
^{FA, FA') ® ^{FA', FA") cfa,fa',fa"' ^(^^' ^^")
Diagram 6.11
Faa
^(FA.FA)
Diagram 6.12
j^{A,A')
j^(A,A')
l0j^{A,A')
j^{A,A')®I
OLA ® GaA'
FaA' ® OLA'
^{FA, GA) ® ^{GA, GA') ^{FA, FA') ® ^{FA', GA')
CFA,GA,GA'
cfa,fa\ga'
^{FA.GA')
Diagram 6.13

6,2 Enriched categories
303
,-1
\oiA0pA
^{FA, HA) ® ^(HA, KA)
\cfa,ha,ka
^{HA.KA)
-1
7®/
O^A ® IHA
^{FA, HA) ® "^{GHA, LHA)
\Gfa,ha ® 1
"^{GFA, GHA) ® ^{GHA, LHA)
\cgfa,gha,lha
^{GFA.LHA)
Diagram 6.14
As expected, one gets the following result.
Proposition 6.2.5 Let i^ be a monoidal category. The small i^-cate-
gories, together with the i^-functors and the i^-natural transformations,
constitute a 2-category written iT-Cat.
Proof If F: si >^ and G: ^ >^ are iT-functors, the composite
GoF', s^ >^ is defined by (G o F){A) = G{F{A)) for Aes/, while
(G o F)aa'^ is the composite
s/{A,A') ^^^' )^(F(A),F(AO) ^^^'^^'^^'^'\'^{GF{A),GF{A')),
liH,K',s^\
\^ are other iT-functors and a: F =^ H, C: H =^ K
are iT-natural transformations, the composite /3oa: F =^ K is defined by
the first composite of diagram 6.14, where A runs through s/. Consider
yet another iT-functor L: ^ >^ and another iT-natural
transformation 7*. G => L, The iT-natural transformation 7* a: G o F => Lo H is
defined by the second composite of diagram 6.14, where A runs through
D
It is routine to check the required commutativities.
Proposition 6.2.6 If "V is a symmetric monoidal closed category, "V
itself can be provided with the structure of a i^-category.
Proof See 6.1.8. D

304 Enriched category theory
s^{A, A!) ^^^' )[FA, FA!\
Gaa'
W.OLA'
{GA,GA\- -^{FA,GA!\
Diagram 6.15
Proposition 6.2.7 Let i^ bea symmetric monoidal closed category and
si a "V-category. Every object A £ s/ induces two "i^-representable^^
i^-functors
s/{A,-):s/ >r, s/{-,A):s/* >^,
whose images under the forgetful functor U of 6.4,4 are just the ordinary
functors on U{si) represented by A.
Proof The object B e s/ is mapped to the object ^{A, B) e "f^ by
s/{A^ —). Given another object C £ s/^ the composite
corresponds by adjunction with a morphism
s^{B, C) > [s^{A, B), si {A, €)]
making si{A, -) a iT-functor.
The case oi si{—^A) is analogous and the statement concerning the
forgetful functor holds just by definition of si{-, A) and si {A, -). D
Finally let us observe that, between 1^-valued 1^-functors, the notion
of 'l^-natural transformations takes a more normal form.
Proposition 6.2.8 Let i^ be a symmetric monoidal closed category.
If si is a i^-category and F, G: .<^ )^ are i^-functors, giving a "V-
natural transformation a: F => G is equivalent to giving a family of
morphisms a a- FA >GA in i^, for A £ si, in such a way that
diagram 6.15 commutes for all objects A, A^ G si.
Proof This reduces immediately to definition 6.2.4 via the
isomorphisms
-^G, [F{A),G{A)]^ ^ 'r(/(8)F(A),G(A)) ^ ^(f(A),G(A)). D

6.2 Enriched categories 305
j/(A, A^) ® ^{B, B') ® s^{A!, A!') ® ^{B\ B")
s^{A, A') ® sd{A!, A!') ® ^{B, B') ® ^{B', B")
\CAA'A" ®CbB'B"
^{A,A")®^{B,B")
{si®^)[{A,B),{A\B"))
Diagram 6.16
We still want to introduce the notion of a ':^-distributor. For this we
must be able to handle 'l^-bifunctors. This will follow from the next
result, where the reader will notice that synmietry is essential.
Proposition 6.2.9 Let i^ be a symmetric monoidal category The
category of small i^-categories and i^-functors is itself provided with the
structure of a symmetric monoidal category
Proof Consider two small iT-categories s/^^. One gets a new i^-
category j?/ 0 ^ by putting:
• \j^®^\ = \j^\ X 1^1;
• {j^®^){{A,B),{A',B')) = s^{A,A')®^{B,B') ioiA.A' e ^ and
B, B' e 3i\
• if A, A!, A!' G si and B, B', B" G ^ the composition map oisi ®^
is given as in diagram 6.16, where the isomorphism is constructed
from the associativity and symmetry isomorphisms;
• if A G J2/, B G ^, the corresponding unit map oi si ®^ is given by
-1
-^I®I
UA ®Ub
^si{A,A)®^{B,B),
The rest is now routine calculations: the unit for the tensor product of
T^-categories is the T^-category J with a single object * and ./(*, *) = /.
n

306 Enriched category theory
IlB',B"e^V'(C^, B") ® ^{B", B') ® 4>(B', A)
u
]Xs^^xP{C,B)®<l>{B,A)
UcA
{i,o<t>){C,A)
Diagram 6.17
Definition 6.2.10 Let i^ be a symmetric monoidal closed category.
By a i^-distributor (p: si o >^ between small i^-categories, we mean
a i^-functor ^*<S>s/ >i^. By a morphism of i^-distributors, we mean
a i^-natural transformation between the corresponding i^-functors.
The previous definition makes sense by 6.2.9 and 6.2.6. But
considering the composite of two distributors requires the cocompleteness of iT,
as in 7.8.2, volume 1.
Proposition 6.2.11 Let i^ be a cocomplete symmetric monoidal
closed category. The small i^-categories, the i^-distributors and their mor-
phisms constitute a bicategory
Proof The only difficult point is to define the composite of two i^-
distributors 0: s/—b-^^ and i/j: ^—e-^^. Thus 0 and i/j are iT-functors
0:^*(8)ja^ >r, ^:^*®^ >r,
which yields in particular morphisms in iT,
(pBAB'A^: ^* (B, BO ® ^(A, A') > [0(B, A), 0(B', A')],
^CBC'B':^*(C,CO®^(B,BO >[^(C,B),^(C',BO],
for A, A' e s^, B,B' e ^, C,C' G ^.
Given A e s/ and C G ^, we define (^o0)(C A) as the coequalizer in
diagram 6.17 (see 7.8.2, volume 1), where u^ v are defined in the following
way. Fixing B\ B" G ^, we consider first the two morphisms
'^B'AB-A'- ^{B\B')^s^{A,A')^(t>{B',A) >(I){B",A'),
^CB^^c^B^' ^{C,B")^<g{C',C)^^{B'\B') >^l^{C',B'\
corresponding by adjunction with (t>B'AB"A' and ipcB^'CB'- The
morphisms w, V are then the unique factorizations through the first coprod-
uct of the composites of diagram 6.18, where sbs^b" are canonical

6.2 Enriched categories
xIj{C, B'') ® ^{B", B') ® ct){B', A)
l®r~^ (8I
^(C, B'') ® ^{B", B')®I® (t){B', A)
1 ® 1 ® t^A ® 1
^(C, B") ® ^{B", B') ® s^{A, A) ® 4>{B\ A)
n®~4>B'AB"A
^(C,B")®(^{B\A)
\sb"
Wb^^^^C,B)®<\>[B,A)
307
^(C, B") ® ^{B\ B') ® (^[B', A)
1®/-^®!
^(C,B") ®I® ^{B\B') ® (t){B', A)
\l®UA®l®l
tPiC B") ® ^(C, C) ® ^{B\ B') ® (t){B', A)
\yCB"CB'
xl^{C,B')®(t>{B',A)
\sb'
]lBe^i^(C,B)®ct>{B,A)
Diagram 6.18
morphisms of the second coproduct and, for simplicity, the associativity
isomorphisms have been omitted.
Given A^A'^si and C, C G ^, it remains to define morphisms
{■^o<l,)^^^,^,:<€*{C,C')®s^{A,A') >[{'4,oct>){C,A),{^ocj>){C',A')].
By adjunction, this is equivalent to constructing a morphism
x: <^{C',C)®s^{A,A') ® (^ o (t>){C,A) >{ip o <t>){C',A').

308 Enriched category theory
"€(€', C) (8) ^(A, A') ® ^{C, B) ® <j){B, A)
ipiC, B) (8) "^(C, C) ® j^{A, A') ® (j){B, A)
ip(C, B) ® <^{C', C) ® / ® / O s/iA, A') (8) <t>{B, A)
l(8)l(8)ttB<8)WB®l<8)l
I'^CBC'b'^^BABA'
i;iC',B)®(t>iB,A')
Ub
l[Be^i^iC',B)®cf>iB,A')
Iqc'A'
(Vo<^)(C',A')
Diagram 6.19
But for every object V e i^^ the functor V <S> — admits [V^ —] as a
right adjoint (see 6.1.4), therefore it preserves coequalizers (see 3.2.2,
volume 1). Thus constructing x is equivalent to constructing
y: ^(C,C)(8)^(A,^0^(Ube^^^^'BHc)>{B, A)) >(^o0)(C', A')
such that you = y o v. Using the fact that each functor F 0 —
preserves coproducts as well (see 3.2.2, volume 1), constructing y reduces
to constructing a suitable family of morphisms
Vb: ^(C', C) (8) j^{A, A') 0 ^(C, B) 0 (t>{B, A)-
-^(^o(/>)(c^AO.
Still avoiding writing the associativity isomorphisms, these are the
composites of diagram 6.19.
The proof is now a list of lengthy but straightforward computations
left to the reader. D

6.3 The enriched Yoneda lemma 309
f"-Nat(F,G)
r
UAe^^iFAGA)
u\ \v
Diagram 6.20
{llA^iFA, GA)) ® s/{A\ A'O s/{A\ A'O ® {llA^iFA, GA))
\pA' ® Ga'A" Fa'.A" ® PA"
^{FA\ GA') ® ^(GA', GA'O ^(FA', FA'') ® ^(FA'', GA'')
CfA',GA',GA'' C,FA\FA",GA"
^{FA'.GA") ^{FA'.GA")
Diagram 6.21
6.3 The enriched Yoneda lemma
We enrich first some of the constructions of section 6.2, using
completeness assumptions on i^.
Proposition 6.3.1 Let i^ be a complete symmetric monoidal closed
category Given two i^-categories si^ ^, with si small, the category of
i^-functors si >^ and "V-natural transformations can be provided
with the structure of a i^-category, written i^[si^^].
Proof Given two iT-functors F, G: si >^, we must define the object
T^-Nat(F,G) G 1^ of T^-natural transformations from F to G. It is
given by the equalizer of diagram 6.20, where it remains to define u and
V, Fixing A!^A!' G si^ the composites of diagram 6.21 correspond by
adjunction (and symmetry) with morphisms
n^^(n^), G{A)) \ p(A', A"),^(F(A'), G(A"))]
which we choose a&u^v followed by the projection p^a'.A")- This defines
w, V and the rest is lengthy but routine computation. D
Observe that in the case tT = Set, we have defined T^-Nat(F,G) as

310 Enriched category theory
the set of those families {aa- FA ^GA)j^^^ such that the mappings
^{A',A") >^{FiA'),GiA")), f ^ Gif)oaA',
^{A',A") >^{FiA'),GiA")), f^aA"oFif)
coincide for all choices A\A^^es/. This indeed defines the set of natural
transformations from F to G.
Corollary 6.3.2 Let i^ be a complete symmetric monoidal closed
category. The category of small i^-categories and i^-functors is itself
provided with the structure of a symmetric monoidal closed category
Proof The symmetric monoidal structure has been described in 6.2.9
while the closedness follows from 6.3.1. D
When ir is a complete and cocomplete symmetric monoidal closed
category, putting together propositions 6.2.11 and 6.3.1 yields that small i^-
categories, iT-distributors and morphisms of iT-distributors constitute
what should be called a iT-bicategory, a gadget whose precise definition
is left to the reader.
The next lemma, together with 6.2.8, will simpUfy a good deal various
proofs involving objects of iT-natural transformations.
Lemma 6.3.3 Let i^ he a complete symmetric monoidal closed
category. Given a small i^-category s/ and two i^-functors F, G: s/ ^i^,
an object N £ i^ is isomorphic to the object of i^-natural
transformations ir-Nat(F, G) if and only if, for every object V e i^, there is a
bijective correspondence between
A) the morphisms V >N,
B) the i^-natural transformations F => [V,G—].
Proof With the notation of 6.3.1, a morphism /: V >ir-Nat(F, G) is
induced by a morphism f: V >]J^^^[F(A),G(A)] equalizing u,v,
thus by a family of morphisms /a: V > [F{A),G{A)]. This is
equivalent to giving morphisms V 0 F{A) >G{A) and thus, by symmetry
and adjunction, to giving morphisms qa'- F{A) > [V, G{A)]. By 6.2.8,
the iT-naturality of the family {gA)Aej^ reduces to the commutativity of
diagram 6.22, where the first vertical morphism, expressing the action of
the composite iT-functor [F, G—]: si ^'V^ corresponds by adjunction
and symmetry with
[F,GA\(g)^(^,AO ^®^^^'^ [y,GA\® \GA,GA!\ ^^>^^>^^S [y,gA\
The required commutativity is thus equivalent by adjunction to that of
diagram 6.23, where the last two arrows correspond by adjunction and

6.3 The enriched Yoneda lemma
^{A,A')^[FA,FA']
311
[V,G]aa'
[[V,GA],[V,GA']\
V
[9A,1]
Diagram 6.22
fA
[^^9A'
[FA,[V,GA^]]
^[FA'.GA']
fA
[FA, GA] >[^{A, AO, [FA, GA^]]
Diagram 6.23
symmetry with the composites of diagram 6.24. But this last commuta-
tivity is itself equivalent to the relation uo f = v o f, which concludes
the proof by 1.9.5 and 1.5.2, volume 1. D
The previous lemma indicates in particular how to define the
object of iT-natural transformations ir-Nat(F, G) between two iT-functors
F,G: .^ \i^^ when si is not necessarily small and i^ is not
necessarily complete.
Definition 6.3.4 Let i^ he a symmetric monoidal closed category.
Given a i^-category s/ and two i^-functors F,G: -^ \^^ an object
Nof'V is called the object of i^-natural transformations from F to G,
and one writes N = ir-Nat(F, G), if for allV ei^ there exist bijections,
natural in the variable V e i^, between
A) the set of morphisms V >N,
B) the class of i^-natural transformations F =^ [V, G—].
Theorem 6.3.5 (Enriched Yoneda lemma)
Let i^ be a symmetric monoidal closed category and si a small "V-
category For every object A e si and every i^-functor F: si >ir,
the object of i^-natural transformations from si {A, —) to F exists and
there is an isomorphism in i^,
Tr-Nat(j^(A, -"), F) ^ F{A),

312 Enriched category theory
[FA, GA] ® j^iA, A') ^{A, A') ® [FA', GA']
\i®Gaa' Faa'®!
[FA, GA] ® [GA, GA'] [FA, FA'] ® [FA', GA']
\cfa,ga,ga' \cfa,fa',ga'
[FA,GA'] [FA,GA']
Diagram 6.24
j^{B, cf^^' ~\^{A, B),s^{A, C)]
[V,F-
[[V,FB],[V,FC]]
WbA]
Diagram 6.25
[l.-^c]
[s^{A,B),[V,FC]]
which is i^-natural both in F and in A.
Proof We refer to 6.3.4. Given an object V £ i^ and a morphism
/: V >F{A)^ we must construct a iT-natural transformation
ip:s/{A,-)-
^[V^F-]^
i.e. a compatible family of morphisms (fB- ^{A^B) > [V^F{B)] for
B in ^. They correspond by adjunction with the composites
^{A, B)^V ^^'^ ® ^) [F{A), F{B)] ® F{A) "^^(^)-^(^^ F{B).
The iT-naturality of (p means the commutativity of diagram 6.25 for
B^C e si\ this is equivalent, by adjunction and symmetry, to the
equality of the composites of diagram 6.26, which holds by iT-functoriality of
F.
Conversely given a iT-natural transformation (^: si {A, -) > [V, F-],
we get the composite
V—"^^-^ [^{A, A), F{A)\
[uaA]
[I,F{A)\^F{A)
where Tpj^ corresponds with (pA by symmetry and adjunction.
It is now straightforward to show from the definitions that we have
defined reciprocal and natural constructions. D

s^{A,B)®si{B,C)®V
\FAC®f
[FA,FC]®FA
FC
6.4 Change of base 313
j^{A,B)®F®j^{B,C)
LPab®/® 1
[FA,FB]®FA®[FB,FC]
\^'^FA,FB ® 1
FB®[FB,FC]
[FB,FC]®FB
l^yFA.FC
FC
Diagram 6.26
Corollary 6.3.6 Let i^ be a complete symmetric monoidal closed
category. For every small i^-category si ^ the mapping
can be extended to a i^-functor called the i^-Yoneda-embedding.
Proof It remains to construct, for A^A^Es/
But by the i^-Yoneda-lemma (see 6.3.4), we have indeed such an
isomorphism Yaa' • Checking the axioms is now routine. D
6.4 Change of base
We study the effect of a "morphism of monoidal categories" on the
corresponding enriched categories, functors and natural transformations. As
a special case, we obtain the ordinary categories, functors, ... associated
with enriched ones.
-^if^ of monoidal categories con-
Definition 6.4.1 A morphism F:
insists in the following data:
A) a functor F: iT ^tT;
{2) for each pair {A, B) of objects of i^, a morphism of iV^,
TAB' F{A)®F{B) >F{A®By,

314 Enriched category theory
{FA ® FB) ® FC""^^'^^'^? FA ® {FB ® FC)
rAB®C
F{A ®B)®FC
TA®B,C
l®rBC
FA ® F{B ® C)
TA,B(S>C
F({A®B)®C) — ^ F(A®{B®C))
F[aABc)
Diagram 6.27
TIA
TAL
FI® FA^^^F{I®A) FA® FI^^^F{A® I)
e®l
J®FA-
FIa l®e
FrA
IpA
-> FA
FA®J-
rFA
^ FA
Diagram 6.28
C) a morphism e: J ^F{I) of iT, where I stands for the unit of'V
and J for the unit of iT.
These data must satisfy the following conditions:
A) the morphisms tab ^^^ natural in A, B;
B) diagram 6.27 commutes for all objects A^B^C of i^ (associativity
condition);
C) both parts of diagram 6.28 commute for each object A of i^ (unit
conditions).
When 1^, T^ are symmetric monoidal categories, F is a morphism of
symmetric monoidal categories when moreover
D) diagram 6.29 commutes, for every pair A, B of objects of "V
(symmetry condition).
Proposition 6.4.2 For every monoidal category i^, the forgetful tunc-
tor i^{I^'-)\ tT >Set is a morphism of monoidal categories, when Set
is provided with its structure of cartesian closed category. If i^ is
symmetric, this is a morphism of symmetric monoidal categories.

6.4 Change of base
FA (8) FB ^^^'^^>FB ® FA
315
TAB
tba
FiA ® B)
^F{B ® A)
FSAB
Diagram 6.29
F{s^{A,B)) ®F{s^{B,C))
F{s^{A,B)®s^{B,C))
\Fcabc
F{s^{A,C))
Diagram 6.30
Proof Given morphisms a: / >A and b: I >B in iT, the
composite
.-1
-^A ^—>A®I-
\®b
^A®B
can be written TAB^CL-tb)^ which yields a mapping
TA^B'^ r{i,A) X r{i,B) >r{i,A®B),
On the other hand the identity on / yields a mapping
It is routine to check the conditions of 6.4.1.
D
Proposition 6.4.3 Let F: 'V >ii^ be a morphism of monoidal
categories. F induces a 2-functor (p\ iT-Cat ^^^-Cat.
Proof Let js/ be a iT-category. The '^^-category (f{s/) is defined in the
following way, with the notations of 6.4.1:
• the objects of (p{s/) are those of j?/;
• for A.Bej^, ^\j^){A,B) = F{j^{A,B));
• for A^B^Ce j/, the corresponding composition morphism of (p{^)
is the composite of diagram 6.30;

316 Enriched category theory
• for A G s/^ the corresponding unit morphism of (p{s/) is the
following composite:
J ^ >F{I) ^^^-^^F(^(AA)).
If G: si >^ is a iT-functor, the 'T'-functor (^(G): (^(ja^) >^[0)
has the same action as G on the objects, while the morphism (^(G))^^,,
for A, A! e si, is just F(GaaO-
If if: si >^ is another iT-functor and a: G => if is a iT-natural
transformation, we get a '^^-natural transformation (^(a): (^(G) => (^(if)
by defining ^(q^)a as the following composite, for A^ si\
.F{I)^^^^)f{s^{G{A\E{A))).
The rest is routine computations. D
Corollary 6.4.4 For every monoidal category there exists a ^^forgetfuF
2-functor
U: ^-Cat >Cat.
Proof Just apply 6.4.3 to ir(/, -); see 6.4.2. D
If j/ is a iT-category, the category U{si) obtained from 6.4.4 is often
referred to as the ordinary category underlying si. One has:
• \U{^)\ =1^1;
• the composite of /: / >si{A,B), g: I >si{B,C) is given by
I^^ >I ® I ^®^ >si{A, B) ® si{B, C) ^-^^^ ) si {A, C).
Observe also that a given /: A >B in U{si) yields a morphism
si{XJ): s^[X,A) >si{X,B)
in i^: this is just the composite
j^{X,A)^j^{X,A)®I ^®f )j/(X,A)®si{A,B) ""^^^ >jaf(X,B),
An analogous definition holds for si{f,X).
Proposition 6.4.5 Let F: i^ >i(^ be a morphism of symmetric
monoidal categories, where i^ and iV^ are in fact symmetric monoidal closed
categories. Under these conditions there exist morphisms in H^,
(Tab: F[A,B] >[F{A),F{B)l
for every pair A, B of objects of i^, these morphisms satisfying the
following conditions:

6.4 Change of base
F[[A, B] ® [B, C]] ?^^^^—> F[A, C]
'^[A,B],[B,C]
317
F[A,B]®F[B,C]
(Tab ^ (tbc
(TAG
[FA,FB]®[FB,FC]
FI
cfa,fb,fc
Diagram 6.31
FUA
t [FA.FC]
^F[A,A]
CTAA
Ufa
Diagram 6.32
^[FA^FA]
A) the morphisms gab a^e natural in A^B;
B) diagram 6.31 commutes, for all objects A^B^C of t^ (composition
condition);
C) diagram 6.32 commutes, for every object A of i^ (unit condition)
Proof The ctab correspond by adjunction with the following composite
F[A,B] ® F{A) ^'^-^'-^ >F[[A,B] ® A] J^^^^!^Ab1^f{B).
The rest is routine computation. D
Proposition 6.4.6 Let i^ be a symmetric monoidal closed category
with coproducts. The functor
r{!,-): r >Set
has a left adjoint F: Set >i^. This adjoint functor is itself a mor-
phism of symmetric monoidal categories with the additional fact that
the required natural morphisms
tab: F{A) ® F{B) >F{AxB), e: I >F{1)
are isomorphisms.

318 Enriched category theory
Proof For a set A, define F{A) = Ua^' ^^^ ^"^"^ copower of /. By
3.2.2, volume 1, there are bijections
^ (Ua^' ^) - Ua^^^^ ^) - ^^^(^' '^(^' ^))-
Observe that by definition, F(l) = jj^l = /, yielding the required
isomorphism s.
Since tensoring with an object of i^ preserves coproducts (see 3.2.2,
volume 1), we have, for all sets A, jB,
F{AHF{B)^(jlj)^[]l^l)
ULaxB -LiAxB
^ F(A X jB),
yielding the isomorphism tab-
The rest is now routine calculations. D
Proposition 6.4.7 Let i^ be a symmetric monoidal closed category
with coproducts. The forgetful functor
U: ^-Cat >Cat
of 6.4.4 has a left adjoint.
Proof If ^ is a small category, define a l^-category ^ as follows:
• l5l = \n
• ^(A, B) = F{'^{A, jB)), where F is left adjoint to ir(/, -) (see 6.4.6)
and A, jB G ^;
• CABC' ^{A, B) (8)¥(jB, C) >^(A, C), for A, jB, C G ^, is the
composite
F(^(A, B)) ®F{^{B, C)) ^F(^(A, B)x^{B, C)) >F{^{A, C))
where the isomorphism is that of 6.4.6 and the arrow is the image
under F of the ordinary composition in ^;
• ua'- I >^{A,A), ioT Ae^ is the composite
I^F{1) >F{^{A,A))
where the isomorphism is that of 6.4.6 and the arrow is the image
under F of the mapping applying * g 1 on 1^.
It is just routine to check that V is indeed a T^-category.
It is easy to define a functor 7: ^ >U(^:

6.4 Change of base 319
• 7(C) = C for C € ^;
• if /: A >B is a morphism of ^, 7(/) is the arrow
sf. I >TT i^F{i).
The definitions of cabc and ua imply immediately that 7 is a functor.
Now given a functor G: ^ >C/(j/), where j/ is a iT-category (not
necessarily small), we get the required unique 1^-factorization G:
^ >s/ in the following way:
• G{C) = G{C) for Ce^',
• Gab: ^(^, B) > j/(G(A), G(jB)) must thus be a morphism of the
type F{^[A,B)) >j/(G(A),G(jB)); it is the one corresponding
by adjunction with the action of G,
^{A,B) >u(s^{G{A\G{B))) ^r{l,s^{G{A\G{B))).
The details follow immediately from 6.4.6. D
Proposition 6.4.8 Let i^ he a symmetric monoidal closed category
with coproducts. Consider a small category ^ and the corresponding
free i^-category ^ (see 6.4.7). For every i^-category j/, there is an
isomorphism of categories between
• the category of functors and natural transformations from ^ to
• the category of i^-functors and i^-natural transformations from ^
to s/.
Proof The last part of the proof of 6.4.7 has been developed for an
arbitrary 1^-category j/. This already yields a bijective correspondence
between
• the functors G: ^ >U{s/),
• the iT-functors G: ^ >s/.
We must prove now that given functors G,H: ^ > TLoi/ with
corresponding tT-functors G, H: ^ \si, there is a bijective correspondence
between
• the natural transformations G ^ H^
• the T^-natural transformations G =^Tl.

320 Enriched category theory
c^{C,D) ^^^>Uj^{GC,GD)
HcD
Us^{\,aD)
Diagram 6.33
A natural transformation a: G =^ iif is by definition a family of mor-
phisms ac'- G{C) ^H{C) in C/j/, thus by 6.2.4 a family of morphisms
ac' I > j/(G(C), H{C)) in 1^, with diagram 6.33 commutative for all
C, D in ^. But since
C/(^(G(C), G{D))) ^ r(l, ^(G(C), G{D))),
we can transform the diagram by the adjunction of 6.4.6 and get equiva-
lently the commutativity of diagram 6.34, which is precisely the
requirement for a: G =^ H being iT-natural (see 6.2.4). D
6.5 Tensors and cotensors
Definition 6.5.1 Let i^ be a symmetric monoidal closed category si
a i^'Category and A e si,V ^ i^ two objects.
• The cotensor of V and A exists if there is an object [V, A] E si
together with isomorphisms
j^{B,[V,A])^[V,j^{B,A)]
in 'V which are i^-natural in B e si. We say si is cotensored when
[V, A] exists for all objects V ei^, Aesi.
• The tensor ofV and A exists if there is an object V^A^ si together
with isomorphisms
si{V^A,B)^ [V,j^{A,B)]
in 'V which are i^-natural in B e si. We say si is tensored when
V<SiA exists for all objects V ei^^ Aesf.
Example 6.5.2
In the case i^ = Set and si an ordinary category with products and
coproducts, one has

6.5 Tensors and cotensors
j^(GC,GD)
321
'^{C,D)
Hcd\
s^(HC,HD)
s^{GC,GD)®I
l®aD
s^{GC, GD) ® j^{GD, HD)
GC,GD,HD
I®s/{HC,HD)
s/{GC,HD)
^GC,HC,HD
ac®l
s/ {GC, He) ® s/ (HC, HD)
Diagram 6.34
•^ (Ilie/^' ^) - nie,-^(^' ^) - Set [/, ^{A, B)],
from which one concludes that the cotensor [/, A] is just the /-th power
of A and the tensor I <S> A is the /-th copower of A, for A, jB G ^ and
/ G Set.
The following proposition indicates in particular that no notational
confusion at all arises when j/ = 1^ in 6.5.1.
Proposition 6.5.3 Let i^ be a symmetric monoidal closed category
Then i^ is both a tensored and cotensored 'V-category, with the tensor
V ® A and the cotensor [V, A] as in 6.1.7.
Proof Indeed the isomorphism
[V^A,B]^[V,[A,B]]

322 Enriched category theory
is equivalent to the isomorphisms
r(x, [v^A,B])^r[x, [V, [A,b]])
for every X E i^ (see 1.9.5 and 1.5.2, volume 1) and this holds since
r{x, [V (g) A,B]) ^ r{x ®v®A,B)^ r{x ® v, [a,b])
^r(x,[v,[A.B]]).
From this we deduce immediately that
[B,[V,A]] ^ [B^V.A] ^ [V^B.A] ^ [V,[B,A]]. D
Proposition 6.5.4 Let i^ be a symmetric monoidal closed category, s/
a "r-category A^B e ^ and V^Wei^.If^is cotensored,
j^i^B, [V,[W,A]])^s^(^B, [V®W,A])^j^[b, [W,[V,A]]'),
and if ^ is tensored,
j^{V ®{W^ A),B) ^ s^{(y ^W)^A,B)^ s/{W 0 (V 0 A), B).
Proof For every object B e s/,
^(b, [V,[V^,A]]) - [V,^(B,[V^,A])]
^[V,[W,s/{B,A)]]
^ [V^W,j^{B,A)]
^j^{B,[V^W,A])
from which follows the first statement.
An analogous proof holds for tensors. D
Proposition 6.5.5 Let i^ bea symmetric monoidal closed category and
V E i^. Let J2/ be a i^-category. If s^ is cotensored, the correspondence
[V,-]:j^ >s/, A^[V,A]
induces a i^-functor and if si is tensored, the correspondence
V®-'.si >si, A^V®A
induces a i^-functor as well.
Proof Suppose si is cotensored. For A^Bes/, the composite
jU^iA^ [.^(A,B),^{A,B)] - ^(a, [j^{A,B),B])
induces a morphism ev^B^ A > [j2/{A,B),B] in U{si). The 'V-hmc-
toriality of [Vi ~] is then expressed by morphisms
^(A,B) >s^{[V,AUV,B])

6.5 Tensors and cotensors
323
^i[V,A],[V,A]) I
[V,j/([V,A],A)] j^{V®B,V®B)
[1,^A, evvA)] L/(l®evAB,l)
[V, j^{[V, A], [j^{A, B), B])] s^{V ® (j/(A, B) ® A),V ® B)
[V, [^(vl, B), ^([F, A], B)]] j/(^(A, B) (g) (y ® A), V®E)
[^{A,B), [V,^([V,A],B)]] [s/{A,B),^{V® A,V® B)\
[^(Afi),^(
Diagram 6.35
corresponding by adjunction with the first composites of diagram 6.35.
In the same way the l^-functoriahty of V 0 — is expressed by mor-
phisms
which correspond by adjunction with the second composites of
diagram 6.35 (see 6.5.2). D
Proposition 6.5.6 Let i^ be a symmetric monoidal closed category If
s^ is a tensored and cotensored i^-category, for every V E i^ the two
i^-functors
[V,-]:j/ >j/, V(8)-:j/ >j/
induce isomorphisms
s/{V^A,B) ^s/{A,[V,B])
which are i^-natural in A,Be^.
Proof For every W ei^, (see 6.5.12),
[W, s^iy (8) A, B)\ ^ j/(W^ (8) (F (8) A), B) ^ ^{V 0 (W 0 A), B)

324 Enriched category theory
^ [V, j^{W ® A, B)] ^j^{W0A, [V, B])
^[W,J^{A,[V,B])],
from which
by applying 1^(/, —). One concludes by 1.9.5 and 1.5.2, volume 1. D
Proposition 6.5.7 Let i^ be a complete symmetric monoidal closed
category. For every small i^-category s/, the i^-category of i^-functors
from s/ toi^ and i^-natural transformations is both tensored and coten-
sored.
Proof Let F: si >ir be a iT-functor and V e i^. It suffices to
define
V^F:s/ >j/, A^V^F{A)
with structural morphisms
^(A B) ?AB_^ [F{A), F{B)] (^^~H [V 0 F{A), V 0 F{B)];
see 6.5.5. In an analogous way
[V,F]:j^ >r, A^[V,F{A)]
has the structural morphisms (see 6.5.5)
^{A,B) ^^^ y[F{A),F{B)] ^^^-^^^^[[V,F{A)],[V,FiB)]\.
It is routine to check that those definitions fit our needs (see 6.3.3). D
The notion of cotensor allows in particular a generalization of 6.3.3.
Lemma 6.5.8 Let i^ a complete symmetric monoidal closed category
s/ a small i^-category and ^ a cotensored "V^-category. Given two i^-
functors F, G: .c/ > .^; an object N e i^ is isomorphic to the object of
i^-natural transformations Tr-Nat(F, G) if and only if, for every object
V £ "T, there is a bijective correspondence between
A) the morphisnis V >N,
B) the i^-natural transformations F => [V, G-].
Proof The proof of 6.3.3 can almost be repeated. With the notsr
tion of 6.3.1, a morphism /: V >Tr-Nat(F, G) is induced by a mor-
phism V ^nA€j/^(-^(^)'^(^))' ^^^^ equivalently by a family of

6.6 Weighted limits 325
morphisms /a- V >^[F{A)^G{A)). By cotensorization of ^, this is
equivalent to giving a compatible family of morphisms
I-
[V,^(F(A),G(A))] ^^(f(A), [V,G(A)]),
thus finally to giving a iT-natural transformation F => [V,G—]. One
concludes again by 1.9.5 and 1.5.2, volume 1. D
6.6 Weighted limits
Enriching the notion of limit with respect to some base category i^ is
much more subtle than the straightforward generalizations of section 6.2.
Indeed, given a functor F: si >^, the limit of F is defined using the
notion of "cone on F": this is a pair (jB, a) where jB G ^ is an object and
a\ Ab => F is a natural transformation, with Ab the constant functor
on the object jB, i.e. the composite
j/ >1 ^^—>^
where 1 is the terminal category (one single object * and just the identity
on it) and ^b is defined by 5b(*) = -B.
When i^ is, for example, a symmetric monoidal closed category and
F is a iT-functor, for every object jB G ^ a 1^-functor 5b- ^ >^ is
easily defined on the unit l^-category J> (see 6.2.10): just put 5b(*) = -B,
where * is the unique object of o^, and choose the morphism
^(*, *) = /—m—^^(B, B)
as Eb)**- But there is no way to find a canonical 1^-functor j/ >^,
except when / G 1^ is the terminal object (for example, when 1^ is
cartesian closed; see 6.1.6).
The first observation for enriching the notion of limit is the fact that in
many fundamental results of the previous chapters, we had to deal with
a situation like in diagram 6.36, where F, G are functors and Elts(G) is
the category of elements of G\ the limit considered was that oi F o cpQ
(see 2.15.6, 3.3.1, 3.7.2, ..., volume 1). Restricting one's attention to
limits of the form lim(F o (J)q) is not at all a restriction since, choosing
for G the constant functor on the singleton, the category of elements of
G is just j/ itself and ^^r = 1^, so that we recapture the Hmit of F.
But the key observation for enriching the notion of limit is the
following fact

326 Enriched category theory
Elts(G)-^^ - ^
Diagram 6.36
Lemma 6.6.1 In the situation which has just been described, there
exist natural bijections in B e ^,
Nat(AB, Fo(J)g) = Nat(G, ^{B, F-))
where A^: Elts(G) >^ is the constant functor on the object B ^ 3S.
Proof Given a cone (9(A,a)- B >(Fo (f)c)[A^a)).^.^ we define a
natural transformation a: G ^ ^{B,F-) by putting ^^(a) = q{A,a)'>
for a G G{A)\ it is natural since given /: (A,a) >{A'^a') in Elts(G)
^(B, F(/)) o aA{a) = F{f) o g(^,,,,) = a^^(a') = a^^ o G(/)(a).
Now, given a natural transformation a: G =^ 3S{B^F—), we define
9(A,a)- -S >F[A) by putting 9(A,a) = ola{p)' This is a cone on Focpc
since given /: {A, a) >(A',a') in Elts(G), the naturality of a implies
F{f) ^ QiA,a) = F{f) o aA = ^{B, F{f)) o aA{a)
= aA' o G(/)(a) = aA'(a') = q^A^a')-
The fact that we have defined reciprocal bijections is obvious from the
definitions. The naturality in B is immediate since, given b: B' >B
and the previous cone g(A,a)? the composite cone is 9(A,a) ^ & which is
applied on /?: G => ^[B', F-) defined by
pA{a) = q^A,a) ob = aA{a) ob = ^F, F{A)) o aA{a), D
Corollary 6.6.2 In the situation of the beginning of this section, the
hmit of the functor F o (J)q exists iff there is an object L e ^ together
with bijections, natural in B e ^,
Nat(G,^(jB,F-)) ^^{B,L).
Proof Just apply 6.6.1 and the fact that the limit exists when there is
L E 38 and natural bijections
Nat(AB,Fo0G) = ^(B,L);

6.6 Weighted limits 327
see 3.2.3, volume 1. D
Choosing B = Lin the previous formula yields the "canonical" natural
transformation A: G => ^{L,F—) associated with the limit: it has the
universal property that given /jl: G =^ SS[B^F—)^ there exists a unique
morphism 6: B >L such that /x = ^F, F—) o A.
Definition 6.6.3 Let i^ be a symmetric monoidal closed category.
Given i^-functors F: s^ >^, G\ s^ >ir, the i^-limit of F
weighted by G exists when:
A) for every B e ^, the object i^-NBt{G,^{B,F-)) of i^-natural
transformations exists;
B) there exists an object L E ^ and isomorphisms in i^,
Xb: ^-Nat(G,^(B,F-)) ^^(B,L),
which are i^-natural in B.
We write in general WmcF for this weighted limit, when it exists. When
limci^ exists for all choices of F, G, with si small, Sd is said to be
incomplete.
As proved in 6.3.1, when si is small and i^ is complete, the required
objects of 1^-natural transformations exist in any case.
The 1^-naturality in jB G ^ is worth a comment. We have a first i^-
functor ^(—, Ly. ^* > i^ and, when si is small, a second one obtained
as the composite
where Y{B) ^ ^(B,-) is the -T-Yoneda-embedding (see 6.3.4), F* is
composition with F and (G, —) is the iT-functor represented by G. What
is required is the existence of a l^-natural isomorphism A between those
two iT-functors. Applying 6.2.8, this can be expressed by isomorphisms
Ab as in 6.6.3 and the commutativity of some diagrams. Due to the first
requirement in 6.6.3, these last commutativities make sense even when
si is not small, and it is what we mean by the l^-naturality in jB G ^.
The notion of weighted iT-colimit is just dual ... which implies in
particular that the weighting f^-functor G is now contravariant!
Definition 6.6.4 Let i^ be a symmetric monoidal closed category.
Given i^-functors F: si >M and G: si* ^V, the i^-colimit of F
weighted by G exists when:
A) for every B G ^, the object ')r-Nat(G,^(F-, B)) of i^-natural
transformations exists;

328 Enriched category theory
j/(C, A) (8) j/(A, B) (8) j^{B, D)
1 <8) cabd
\CCAD
Diagram 6.37
B) there exists an object L G ^ and isomorphisms in i^,
which are i^-natural in B.
We write in general coHm^i^ for this weighted colimit, when it exists.
When coHmci^ exists for all choices of F^ G, with si small, ^ is said to
be i^-cocomplete.
In order to specify two important particular cases of weighted Hmits,
we need some lemmas related with the tensor product of iT-categories,
as described in 6.2.9.
Lemma 6.6.5 Let i^ be a symmetric monoidal closed category For
every small 'V-category si there is a 'V-functor
si\ j/* (g) si >ir
with values s^{A,B) on the objects.
Proof Given objects {A, B), (C, D) G j/* 0 si, it remains to define the
required morphism
^abcd: ^*(A, C) 0 si{B, D) > [si{A, B), si(C, D)\,
By adjunction and symmetry, this corresponds with the composite of
diagram 6.37. The rest is routine computation. D
Lemma 6.6.6 Let i^ be a symmetric monoidal closed category. The
unit 'V-category J is i^-isomorphic to its i^-dual.
Proof Indeed \J\ = {*} = \J\* and ^(*, *) = / = ^*(*, *), from which
we get the result. D
Lemma 6.6.7 Let i^ be a symmetric monoidal closed category, M a
i^-category and J the unit i^-category. The category of i^-functors
J ^^ and i^-natural transformations can be provided with the
structure of a y-category, i^-isomorphic to Si,

6.6 Weighted limits 329
Proof A iT-functor F: J >^ is the choice of an object F(*) G ^
together with a morphism (/?: ^(*, *) >^(F(*), F(*)). But the second
axiom for a iT-functor implies that (/? = tiF(*) ? so that giving a iT-functor
F: ^ >^ is equivalent to giving an object F(*) G ^.
Next considering the construction of ir-Nat(F, G) in 6.3.1 we
realize that both products reduce to a single factor since ^ is a singleton.
Moreover j/(A',A'') = ^(*,*) = /, from which u and v are just the
identity on ^(F*, G*). In particular the equalizer k exists and it is the
identity on ^(F(*),G(*)). Thus ^-Nat(F,G) ^ ^(F(*),G(*)),
yielding the conclusion. D
With the notations of 6.6.7, we write Sb'- ^ >^ for the iT-functor
corresponding with the object jB G ^. By 6.6.6, Sb is both iT-covariant
and 1^-contravariant.
Definition 6.6.8 Let i^ be a symmetric monoidal closed category and
j/, Sd two i^-categories.
• By the end f^^^F{A, A) of a i^-functor F: s^^'^si >^, we mean
the i^-limit ofF weighted by s/: s/* <S>s/ >i^, when this exists.
• By the coend /^^'^F(A, A) of a i^-functor F: j/ 0 j/* > J^, we
mean the i^-colimit ofF weighted by si\ s^*<^s^ >1^, when this
exists.
Proposition 6.6.9 Let i^ be a symmetric monoidal closed category
and j/ a i^-category. Given i^-functors F,G: s^ZHX^i ^^^ following
conditions are equivalent:
A) the object of i^-natural transformations ir-Nat(F, G) exists;
B) the end of the i^-functor
T:j^*®s/ >r, {A,B)^[F{A),G{B)]
exists;
C) the weighted i^-limit HmirG exists.
When these conditions are realized,
^-Nat(F, G)= [ T{A, A) = limpG.
J AGs/
Proof Observe first that T organizes itself into a iT-functor. The
structural morphisms
(^* X ^) {{A, B), {C, D)) . [[F{A), G{B)], [F(C), 0@)]]
correspond by adjunction with the composites of diagram 6.38, where
the second arrow itself corresponds by adjunction and symmetry with
the composite of diagram 6.39.

330 Enriched category theory
s^{C,A)®^{B,D)
IFca^Gbd
[FC,FA]®[GB,GD]
[[FA,GB],[FC,GD]]
Diagram 6.38
[FC, FA] 0 [FA, GB] 0 [GB, GD] 0 FC
\cfc,fa,gb 0 1 0 1
[FC, GB] 0 [GB, GD] 0 FC
\cfc,gb,gd ^ 1
[FC,GD]^FC
GD
Diagram 6.39
For each V e i^, giving a iT-natural transformation a: s/
is giving a family of morphisms
[V,T-]
aAB-s^{A,B)-
\V,[F{A),G{B)\]
which is iT-natural in A, B\ see 6.2.8. On the other hand giving a i^-
natural transformation /3: F =^ [V, G—] is giving a family of morphisms
Pc:F{C) >[V,G{C)]
which is iT-natural in C.
Given a, the composite
I ^^ )^(C,C) ""^^ )[V, [F(C),G(C)]]
defines a morphism V > [F(C),G(C)] in tT which, by adjunction,
corresponds with a morphism V 0 F{C) >G{C) and finally with a
morphism /3c- FC >[V, GC]. Conversely given /3, one constructs aAB
as the composite in diagram 6.40.

6.6 Weighted limits
\Fab
[F{A\F{B)\
|[1,^b]
[f{A\[V,G{B)\]
\V,[F{A),G{B)]]
Diagram 6.40
331
It is immediate that the correspondences a h^ /3, /3 h^ a are reciprocal.
Observe that given W E i^^ this implies also the existence of natural
bijections between
• the iT-natural transformations s/ => [W, [V, T—]],
• the iT-natural transformations F =^ [W, [V, G—]],
just because [W, [V, -]] ^ [l^ 0 V, -]; see 6.5.4. Therefore by 6.3.3 we
get the isomorphism
^-Nat(j/, [V,T-]) ^ ^-Nat(F, [V,G-]).
This concludes the proof of B) <^ C) since the end of T exists when
L J A£j^ J
while the weighted 1^-Iimit limirG exists when
r-Nat(F, [V,G-]) ^ [V^liuiFG],
The case of ir-Nat(F, G) is treated by the same remark as above; it
exists when we have bijections
^-Nat(F,[V,G-]) ^r{V,rMat{F,G)),
Replacing V hy V (S>W yields by 6.5.4 bijections
TT-Nat^F, [W, [V,G-]]^ ^ r(w, [V,^-Nat(F,G)])
and thus by 6.3.4 isomorphisms
Tr-Nat(F, [V, G-]) ^ [V, Tr-Nat(F, G)]. D

332 Enriched category theory
Proposition 6.6.10 Let i^ be a complete symmetric monoidal closed
category si a 'V-category and Aesi,Vei^ two objects.
• The cotensor [V, A] of the objects V, A exists if and only if the i^-
limit of 6a'' ^ ^^^ weighted by by\ J >i^ exists; when this is
the case, both objects are isomorphic.
• The tensor V <S) A of the objects V, A exists if and only if the i^-
colimit of 6a'' ^ >^ weighted by by\ J> >i^ exists; when this
is the case, both objects are isomorphic.
Proof The iT-functor j/(jB,5a-): ^ >^ is just the iT-functor
Ss/{B,A)^ when B e si. Therefore by 6.6.7
r-NBt{6v,^{B,6A)) =r-Nat{6v,6^^B,A)) = [V,si{B,A)].
The existence of the weighted Hmit Hm^^ 6a thus reduces to the
existence oi L E si together with l^-natural isomorphisms [V, j/(jB, A)] =
si{B, L), which is just the definition of the cotensor [V, A]; see 6.5.1. An
analogous proof holds for tensors. D
Proposition 6.6.11 Let i^ be a complete symmetric monoidal closed
category. For every i^-category si and every object A G si:
A) the i^-functor si {A, -): si >i^ preserves all weighted i^-limits;
B) the i^-functor si{-,A): si >ir transforms weighted i^-colimits
into weighted i^-limits.
Proof Consider a small iT-category ^ and two iT-functors F: Sf >si,
G: Q} >i^. Suppose the weighted iT-Hmit WmcF exists and is given
by L G j/ together with iT-natural isomorphisms
^-Nat(G, si{B, F-)) ^ si{B, L)
ioT B e si. For every V G 1^, the isomorphisms (see 6.5.7)
ir.Nat(G, [V,j/(AF-)]) ^ [v,f"-Nat(G,j/(A,i^-))]
show that si {A, L) = limG^(A, F-).
The second assertion holds by duality. D
Proposition 6.6.12 Let i^ be a complete symmetric monoidal closed
category, V an object ofi^ and si a Y'-category. If si is cotensored, the
functor [V, —]: si >si preserves weighted i^-limits. If si is tensored,
the functor V (Si—: si >si preserves weighted i^-colimits.

6.6 Weighted limits 333
Proof Let ^ be a small iT-category and F: Q) >j/, G\ Q) >ir
two 1^-functors. If limci^ exists, we have an object L £ s^ together with
1^-natural isomorphisms
for A G j/. If j/ is tensored, applying 6.5.7 we get
^[V,^(A,L)]
^^(A,[V,L]),
from which [V, L] ^ Hmc[V, F-]. D
With view to proving an existence theorem for weighted iT-Hmits, let
us make clear the relation between ordinary limits and weighted limits.
Lemma 6.6.13 Let i^ be a complete and cocomplete symmetric mon-
oidal closed category. Consider, with the notation of 6.4.7,
• a cotensored i^-category s/,
• a small category ^ and the corresponding free i^-category ^,
• a functor F: ^ > C/(j/) with corresponding i^-functor F: ^ > j/,
• the constant functor A/: ^ >i^ on I E i^ and the corresponding
r-functor ^i: ^ >r.
If, for every Aej^, the functor s/{A, —): C/(j/) > i^ preserves limits,
the following conditions are equivalent:
A) the limit of F exists;
B) the i^-limit of F weighted hy Aj exists.
Moreover, the limit object is the same in both cases.
Proof The iT-limit of F weighted by A/ exists if we can find L e s/
together with l^-natural isomorphisms
r-Nat(A~i,s/{A,F-)) ^s/{A,L)
for A E: s/. By 6.3.3, this is equivalent to having a bijective
correspondence between
• the iT-natural transformations A/ =^ [V,j^{A,F-)],
• the morphisms V >s/{A,L)
for every object V G T^.

334 Enriched category theory
A iT-natural transformation a: A/ => [V, j/(A,F-)], corresponds
with a compatible family of morphisms
as: I^V >j/(AF(B)).
Via the isomorphisms
r(l®V,s/{A,F{B))^^r(l, [V,J^{A,F{B))\\
^U{j^)(^A,[V,F{B)])
this corresponds with a cone
aB--A >[V,FB]
on[V,F-] mU{s/).
On the other hand via the isomorphisms
r{v, j^{A,L)) ^r{i^v,s^{A,L)) ^ r(i, [v,^{a,l)])
S r(l,s^{A, [V,L])) ^ C/(^)(A [V,L])
a morphism V > j/(A, L) in i^ corresponds with sone A > [V, L] in
As a conclusion, the existence of the iT-limit of F weighted by A/ is
equivalent to the existence of natural bijections between
• the cones of vertex A on [V, F—],
• the morphisms A >[V,L] in U{s^),
for all A G j/, V G ir. And since [/, -] is just the identity functor on
J3^, this means finally that the ordinary limit of F exists in C/(j/) and
is preserved by each functor [V, -]: U{s^) >U{s^).
It remains to prove that every functor [V,—]: U{s^) >U{s/)
preserves limits. This follows from the bijections
Nat(AA, [V, FH) ^ Nat(Av, ^(A, F-))
^Tr(F,^(A,L)) ^C/(^)(A,[V,L])
since j?/ is cotensored and s/{A,—): C/(j?/) >i^ preserves limits, by
assumption. D

6.6 Weighted limits 335
^(C, D) :^^^:i^^zk§[^(L, FC),s/{L, FD)]
GcD
[o^cA]
[GC.GD] p->[GC,^(L,FD)]
Diagram 6.41
The reader should convince himself that, in 6.6.12, the assumption
"j/(A,-): U{s/) >ir preserves limits" does not follow from 6.6.11
when j/ is not supposed to be i^-complete.
Theorem 6.6.14 Let i^ bea complete and cocomplete symmetric mon-
oidal closed category. A i^-category si admits all weighted i^-limits iff:
A) s/ is cotensored;
B) the underlying category C/(j/) is complete;
C) the functors s/{A, —): U{s/) >i^ preserve (ordinary) limits.
A i^-category si admits all weighted i^-colimits iff:
A) s/ is tensored;
B) the underlying category U[si) is cocomplete;
C) the functors j/(—,A): C/(j/) >i^ transform (ordinary) colimits
into limits.
Proof We prove the first statement; the second one follows by duality. If
s/ admits all weighted 1^-limits, it is cotensored (see 6.6.10) and C/(j/)
is complete (see 6.6.13). Moreover, using the notations of 6.6.13 together
with 6.6.11,
j/(A, limF) ^ si {A, lim^F)
^ lim^j/(A, F-) ^ lim^j/(A,F-) ^ lim j/(A, F-)
which proves that si{A^ —): U{si) >1^ preserves limits.
Conversely, assume that si is cotensored and U{si) is complete.
Choose a small iT-category & and iT-functors F: Q) >j/,
G\ Q) > iT. For an object A G j/, we shall first construct an object L G
si and a "universal" iT-natural transformation ol\ G ^ si{L^ F—)^ that
is a family of morphisms ao'- G{D) >si[L,F{D)), D e S>, such that
the square in diagram 6.41 commutes (see 6.2.8). But
giving aD' G{D) >si{L,F{D)) in iT is equivalent to giving ^d'-
L >[G{D),F{D)] in C/(j/), by definition of a cotensor. The i^-

336
Enriched category theory
L
[GC,FC]
[l,ev]
[GD,FD]
[ev,l]
[GC, [s/{FC, FD), FD]] [[GC, GD] ® GC, FD]
[s^{FC, FD), [GC, FD]] [[GC, GD], [GC, FD]]
[F,l]\ /[G,l]
[9{C,D),[GC,FD]]
Diagram 6.42
naturahty of a, in terms of the morphisms /3d, means the commutativity
of diagram 6.42 for all objects C,D e S^. Defining (/3d: L > [G{D),
F(D)])^ to be the corresponding limit in U{s/) of the global diagram
obtained when C, D run through ^, we get by adjunction the required
f^-natural transformation a: G =^ j/(I/, F—) with the property that, for
every T^-natural transformation 7: G =^ s/{A,F-) with A e s/, there
exists a unique morphism a: A >L in U{s/) such that 7 = j/(a, l)oa.
In other words, we have got natural bijections
^.Nat(G, j/(A,F-)) ^ C/(j/)(A,L).
Fix now an object V e i^ and, in the previous argument, replace F
by the composite iT-functor [V,F-]: ^ >j/. One now gets a limit
(/3^: L^ > [G{D), [V,F{D)]])^^^ and natural bijections

6.6 Weighted limits 337
Let us prove that L^ ^ [V, L]. In fact the diagram defining L is built up
from a family of pairs
[GC, FC] > [^(C, D), [GC, FD]] < [GD, FD]
and the diagram defining L^ is built up from a family of pairs
[GC, [V, FC]] > [^(C, D), [GC, [V, FD]]] < [GD, [V, FD]].
By definition of cotensors, this last diagram is isomorphic to
[V, [GC, FC]] > [V, [9{C, D), [GC, FD]]] i [V, [GD, FD]]
and it is immediately clear that this is just the diagram obtained from
that defining L after appHcation of [V,—]. To prove the isomorphism
l^v ^ [V,L], it remains thus to prove that [V,—] preserves the limit
defining L.
But the very last argument in the proof of 6.6.12 was precisely that
the functor [V, —] preserves limits as long as every representable functor
j/(A, —): U{s/) >i^ does, which is one of our assumption.
Finally we have got natural bijections
^-Nat(G,j/(A,[V,F-])) ^U{s/){A,[V,L])^r{V,s/{A,L))
and we can conclude by 6.3.3 that there exists an isomorphism in 1^,
^-Nat(G, j/(A F-)) ^ s/{A, L). D
Corollary 6.6.15 Let i^ bea complete and cocomplete symmetric mon-
oidal closed category and j/, ^ two i^-categories, s/ being i^-complete.
A i^-functor F: s/ >^ preserves all weighted i^-limits iff
A) F preserves cotensors,
B) F preserves ordinary limits.
Proof The necessity of the conditions follows immediately from 6.6.10
and 6.6.13. The construction of the weighted 1^-Iimit in the proof of
6.6.14 shows that the conditions are also sufficient. D
Corollary 6.6.16 Let i^ be a complete and cocomplete symmetric
monoidal closed category and s/ a tensored and cotensored category.
Then:
• s/ is i^-complete ifFU{s/) is complete;
• s/ is 'V-cocomplete iff C/(j/) is cocomplete.

338 Enriched category theory
Proof Applying the functor 1^(/, —) to the isomorphism of 6.5.6, we
get for A,B es/ and V e i^
U{j^){V0 A,B) ^U{j^){A,[V,B])
which proves the adjunction V<S>— H [V, —] (see 3.1.5, volume 1) and thus
the facts that [V, -]: C/(j/) > j/ preserves limits and V<S)-: s^ >s^
preserves colimits. Reversing the argument at the end of the proof of
6.6.13 yields, with the same notation,
Nat(Av, ^(A, F-)) ^ Nat(AA, [F, F-])
S C/(^)(A, [F, L]) ^ r (F, ^(A, L))
since the functor [V, —] preserves limits. But this means precisely that
the functor si{A^—)\ U{s/) >i^ preserves limits. In the same way
one proves that j/(—, A): U{s/) >i^ transforms colimits into limits.
D
Let us now generalize 2.15.6, volume 1, to the case of 1^-functors.
First of all observe that
Proposition 6.6.17 Let i^ be a complete and cocomplete
symmetric monoidal closed category. For every small i^-category s/, the i^-
category of i^-functors si >i^ and 'V-natural transformations is
incomplete and i^-cocomplete.
Proof By 6.6.16 and 6.5.7, it suffices to prove that a pointwise limit or
colimit of iT-functors is still a iT-functor. We do it for Umits; the case
of colimits is analogous.
Consider a diagram [Fu)d£9 of 1^-functors and l^-natural
transformations indexed by a small category S). By 6.2.8, it makes sense to define
in the following way a 1^-functor L: j/ >ir. It maps the object A on
\vaiD£9FD{A) while the structural morphisms
j^(A, B) >\\mD^9 [limD€^i^D(^), Fd{B)\
can equivalently be given by morphisms
J3f (A, B) > [limDe^FD(^), limDe^FD(B)]
and these are just the factorizations through the limit of the composites
^{A,B) ^^ ) [Fd{A), Fd{B)] JP^ilU [limoe^FDiA), Fd{B)].
It is straightforward to check that L is indeed the required Umit.

6.6 Weighted limits 339
In the previous argument we have used the fact that the functors
[V,—]: i^ >i^ preserve limits, which is clear since they admit the
functors - <S>V: i^ >ir as left adjoints (see 3.2.2, volume 1). The
proof in the case of colimits requires the fact that the functors [—, V]:
'V >1^ transform colimits into limits. This is the case by 6.6.11 and
6.6.15. D
Theorem 6.6.18 Let i^ he a complete and cocomplete symmetric
monoidal closed category For a small i^-category s/, consider the i^-
Yoneda-embedding
Y:s/* >^-Fun(j/,^), Ah^j/(A,-).
For every i^-functor F: s/ > iT, the isomorphism F = colimiry holds.
Proof For every 1^-functor G: si >1^ we must find an isomorphism
^-Nat(F(-), ^-Nat(y(-), G)) ^ ^-Nat(F, G).
This is an immediate consequence of the iT-Yoneda-lemma (see 6.3.5)
which implies ^-Nat(y(-), G) ^ G. D
Examples 6.6.19
6.6.19.a Choose Y = Cat, the cartesian closed category of small
categories (see 6.1.9.b). A iT-category is just a 2-category (see 7.1.1,
volume 1) and a iT-functor is just a 2-functor (see 7.2.1, volume 1). Given
a 2-functor F: si > J^ with si small, the iT-limit of F weighted by
the constant functor Ai: si >i^ on the terminal object of Cat is just
the 2-limit of F (see 7.4.5, volume 1).
Examples 6.6.20
6.6.20.a Pseudo-limits and lax limits can also be presented as special
cases of weighted Cat-limits. Let us indicate this with an example. In
a 2-category j/, the lax equalizer of two morphisms /,p: A \ B is a
morphism k: K >A together with a 2-cell n: f ok =^ gok^ these data
being 2-universal for these requirements (see 7.6.1, volume 1). Write J
for the weighting category with two objects X, Y and two non-identity
arrows w, v: X Iy-^ F: J >si is the 2-functor defined by F(u) = /,
-^(^) = 9' Write 1 for the category with a single object 0 and just the
identity on it; write 2 for the category with two objects 1,2 and one
single non-identity arrow t: 1 >2. The functor G'. tf >Cat maps
X to 1, F to 2, u to G{u)\ 1 >2 defined by G(w)@) = 1 and i;
on G{v)\ 1 >2 defined by G(v)(^) = 2. For every object C G j/, a
2-natural transformation a: G => si{C, F—) consists in giving

340 Enriched category theory
• a functor ax'- 1 >j/(C, A), i.e. a morphism h: C >A in j/,
• a functor ay: 2 >j/(C, jB), i.e. two morphisms /,m: C ).jB in
j^ and a 2-cell A: / => m,
and these data must satisfy the requirements f o h = l^ g o h = m. In
other words giving a is just giving h: C >A together with a 2-cell
\: f o h =^ g o h. Giving such a 2-universal a is thus indeed giving the
lax equaUzer of /, g.
6.7 Enriched adjunctions
The use of weighted limits and colimits now makes possible the direct
generalization of most results of category theory in the context of
enriched category theory. We shall limit our attention to the case of adjoint
functors and Kan extensions.
Definition 6.7.1 Let i^ be a symmetric monoidal category and j/,^
two 'f-categories. Two i^-functors F: s/ > J*, G: 3d >j/ are 'V-
adjointj G left adjoint to F and F right adjoint to G, when there exist
isomorphisms in i^
s^{G{B\A)^^{B,F{A)),
which are Y'-natural in Ae s/, B e J*.
Proposition 6.7.2 Let i^ be a symmetric monoidal closed category
and F: si >^ a Y-functor. The following conditions are equivalent:
A) F hasa left i^-adjoint G: ^ >j^;
B) for every B e ^, there exists G{B) G si together with isomorphisms
s/{G{B),A)^^{B,F{A))
which are i^-natural in A e si.
Proof We make G a l^-functor by observing that the isomorphism
r(j,j^{G{B),G{B))^ ^ r[l,^{B,FG{B))^
yields a morphism t^b- B >FG{B) in the underlying category C/(J^),
riB corresponding with ug(b) • The composite
^(C,B)-^lii^??sI»^(C,FG(B)) ^ si{G{C),G{B))
provides G with the structure of a TT-functor.
It remains to prove the naturality in B of the isomorphisms
0ab: ^{G{B),A)^a{B,F{A)).

6.7 Enriched adjunctions
j^{GC,GB)®j^{GB,A)
341
3S{C,B)®s^{GB,A)
1®0ab
cgc,gb,a
^{GC, B)
Gab
^(C, B) ® ^{B, FA) ^^^^^^^ ) ^{B, FA)
Diagram 6.43
By 6.2.8 this is equivalent, by adjunction, to the commutativity of
diagram 6.43, which follows immediately from the definition of the i^-
functor structure of G. D
Proposition 6.7.3 Let i^ be a symmetric monoidal closed category If
a i^-functor F: si > J^ has a left i^-adjoint, it preserves all weighted
i^-limits.
Proof Choose two iT-functors H: S) >j/, K: S) >i^ defined on a
iT-category Sf, Suppose the weighted Hmit L = limKH exists, yielding
1^-natural isomorphisms in A G j/,
r-Nat{K,j^{A,H-)) ^ j^{A,L).
Write G for the left adjoint to F.
We must find isomorphisms, 1^-natural in jB G ^,
r-N3t{K,^{B,FH-)) ^^{B,F{L))',
putting A = G{B), this is immediate since ^{B,F-) ^ s^{G{B),H-)
and ^{B,F{L)) ^ s/{G{B),L), D
Proposition 6.7.4 Let i^ be a symmetric monoidal closed category.
• A i^-category s^ is tensored iff every i^-functor j/(A, —): j/ > iT,
for A ^ j2/j has a left i^-adjoint.
• A i^-category si is cotensored iff every i^-functor j/(—,A):
jaf* >Tr, for A e si, has a left i^-adjoint.
Proof This is immediate from 6.5.1 and 6.7.2.
n

342 Enriched category theory
A general relation between weighted limits and enriched adjunctions
can now be established. This is a direct generalization of 3.3.1, volume 1.
Proposition 6.7.5 Let i^ be a symmetric monoidal closed category.
Given a i^-functor F: s/ >^, the following conditions are equivalent:
A) F hasa left V-adjoint G;
B) for every object B e^, the'T-limit G{B) of 1^: j/ >j/ weighted
by ^{B,F—) exists and is preserved by F.
Proof If F has a left iT-adjoint G, it preserves iT-Hmits (see 6.7.2).
Moreover, by the l^-Yoneda-lemma (see 6.3.4)
^-Nat(^(B, F-), j/(A, -)) ^ ^-Nat(j/(G(B), -), s/{A, -))
^s/{A,G{B)).
Thus G{B) = lim^(B,F-)l^-
Conversely the assumptions mean
^-Nat(^(B,F-),j/(A,-)) ^s/{A,G{B))
r-Nat{^{B, F-), ^(C, F-)) ^ ^(C, FG{B))
ioT A e j^ and B^C e ^. In particular
s/{G{B),G{B)) ^^-Nat(^(B,F-),j/(G(B),-)),
^-Nat(^(B, F-),^{B, F-)) ^ ^(B, FG{B))
^^-Nat(j/(G(B),-),^(B,F-)),
with the last isomorphism coming from the iT-Yoneda-lemma. Applying
i^{I, —), the identities on ^(jB, F—) correspond with iT-natural
transformations
a^: ^(B,F-) =^^(G(B),-), /3a: s^{G{B),-) ^ SS{B,F-),
and it is a straightforward matter to check that ^a = (<^a)~^- One
concludes the proof by 6.7.2. D
Proposition 6.7.5 leaves us with the problem of finding sufficient
conditions for the existence of the weighted limits lim^(B,F-)lj3^ defined on
the TT-category j/, which is generally large. In practice, it is often more
efficient to reduce the problem to one of the classical adjoint functor
theorems (see section 3.3, volume 1). This is possible via the following
theorem.

6,7 Enriched adjunctions 343
Theorem 6.7.6 Let i^ be a symmetric monoidal closed category si a
cotensored 'V-category, ^ an arbitrary i^-category and F: si >S^ a
i^-functor. Then F has a left i^-adjoint functor if and only if
A) F preserves cotensors,
B) the underlying functor U{F): U{si) >U{^) has a left adjoint.
Proof If F has a iT-left adjoint G, F preserves cotensors by 6.7.3 and
6.6.10. Moreover applying 1^(/, —) to the isomorphisms
si{G{B),A) ^^{B,F{A))
yields bijections
U{^){G{B),A)^U{^){B,F{A))
proving that the ordinary functor underlying G is left adjoint to the
ordinary functor underlying F,
Conversely, suppose G: U[3S) >U{si) is a functor left adjoint to
the functor U{F): U{si) >C/(^). By 6.7.2, it remains to prove the
existence of isomorphisms
si(G{B),A) ^^(B,F(A))
1^-natural in si. The adjunction G H U[F) yields two natural
transformations r/: lugs =^ U{F)oG and e: GoU{F) =^ 1^; see 3.1.5, volume 1.
In particular we have a composite, l^-natural in A,
si{G{B),A) ^^(-^)>^ )^{FG{B),F{A)) -^fell^^(B, F{A)),
Constructing its inverse
^(B, F{A)) >si{G{B), A)
is equivalent, since j/ is cotensored, to constructing a morphism in U[si)
G{B) >[^(B,F(A)),a].
Applying the adjunction G H U{F), this is still equivalent to
constructing a morphism in U{^)
B >f[^(B,F(A)),a].
Since F preserves cotensors, this reduces to constructing
B >[^{B,FiA)),F{A)\,

344 Enriched category theory
i.e. to constructing a morphism in 1^,
^{B, F{A)) >^(B, F{A)),
which we choose to be the identity.
It is now straightforward computation to verify that we have defined
reciprocal isomorphisms. D
Let us conclude with the case of enriched Kan extensions (see section
3.7, volume 1).
Theorem 6.7.7 Let i^ be a complete symmetric monoidal closed
category Consider a i^-functor F: si >^, between two small 'V-
categories si^ ^, and a i^-cocomplete and cotensored i^-category ^.
Under these conditions, composing with F
yields a i^-functor F* admitting a i^-left-adjoint functor written Lan^.
For a Y'-functor G: si >^, the i^-functor Lan^G is called the left
i^-Kan extension of G along F.
Proof Now F* maps a iT-functor H: ^ >^ to HoF. This yields a
1^-functor with structural morphisms
Fhw • ^-Nat(i7, H') > r-NBt{H oF,H' o F)
defined in the following way. By the iT-Yoneda-lemma
^-Nat([-,M],[-,Ar]) ^ [M,N]
for all M, AT G ir (see 6.3.5), thus applying ir(/, -) indicates that
defining a morphism M >N is equivalent to defining a l^-natural
transformation [—,M] => [—, AT], thus for every object V^ G 1^ a corresponding
morphism [V,M] >\y,N]. In our case, M, AT are objects of natural
transformations so that the problem reduces, by 6.5.8, to defining
mappings
Tr-Nat(//, [V, H'-]) >'r-Nat(// o F, [V, H' o F-]).
But since [V, H^-] o F = [V,H^o F-], this is just composition with F.
The TT-naturality is obvious. This completes the definition of F*.
To prove the existence of a tT-left-adjoint to F*, choose a TT-functor
G: si >^. We must define a TT-functor LsiifG: ^ >^ and tT-
natural isomorphisms
Tr-Nat(LaiiFG,H) ^ Tr.Nat(G,HoF)

6.7 Enriched adjunctions 345
^{F-,B')
\lB'
^(G-, colim^(ir_,BoG^)
Diagram 6.44
where H runs through the iT-functors H: ^ >^; see 6.7.2. Given B in
^, we define (LanFG)(B) as the colimit of G weighted by ^(F-, B), To
make this a l^-functor, it remains to construct the structural morphisms
^{B,B') >^(colim^(F-,B)G,colim^(ir_,BoG^)-
Considering the isomorphisms
r-Nat(^(F-, B'),^{G-, colim^(F-,BoG))
= ^(colim^(ir-,BoG^' colim^(ir_,B')G^)
and applying 1^G,—), the identity on the right-hand side corresponds
with a 1^-natural transformation
7B': ^[F-,B') ^ ^(G-,colim^(F-,BoG)-
On the other hand, by 6.6.11
^(colim^(ir_,B)G,colim^(ir_,BoG^)
^lim^(ir_,B)'^(G-,colim^(ir_,BoG^)-
By definition of a weighted limit, constructing the required structural
morphisms is thus equivalent to constructing l^-natural transformations
^(F-,B) ^ [^[B,B'),^{G-,co\\m^^F-^B')G)\
These correspond by adjunction with the composites of diagram 6.44.
The reader will check that this indeed provides Lan^G with the structure
of a 'i^-functor.
It remains to prove the isomorphism
Tr-Nat(LaiiFG, H) ^ ^-Nat(G, H o F).

346 Enriched category theory
^{F-,B)
H
<^{HF-,HB)
\[V,-]
<^{[V,HF-],[V,HB])
k(Al)
nG,[V,HB])
Diagram 6.45
The argument at the beginning of the proof shows that this is equivalent
to finding natural bijections
^-Nat(LanFG, [V, H-]) ^ ^-Nat(G, [V, H o F-])
for V G 1^. To do this, observe that the composites
/ ^^^ >^{F{A), F{A)) (^^^)^ >^(G(A), (LauirG o F){A))
define a l^-natural transformation 7: G =^ Lan^G o F, Given another
natural transformation a: Lan^G => [V, H—] we get the composite
G ^—>LanFG o F ^ * ^ ) [V, H-]oF^ [V, H o F-].
Now given l3: G => [V,H o F—]^ the definition of Lan^G yields
isomorphisms
Tr-Nat(^(F-,B),^{G'',C)) ^ ^((LanirG)(B),G).
Putting G = [V, H{B)], the composite of diagram 6.45 defines a i^-
natural transformation ^{F-,B) ^ ^(G, \y,H{B)\), thus by the
previous isomorphism a morphism
>^({h^nFG){Bl[V,H{B)\).
These last morphisms, for all 6 G ^, define the required T^-natural
transformation LanfrG => [V, H—] which is required and we again leave
to the reader the verification that we have indeed defined reciprocal
bijections. D

6.8 Exercises 347
6.8 Exercises
For simplicity, in these exercises, i^ always denotes a complete and co-
complete symmetric monoidal closed category.
6.8.1 Let j/, ^ be small iT-categories and T: (j/(8)^)*(8)(j/(8) J') > iT
a iT-functor. Prove the "Fubini formula"
f I T{A,B,A,B)^ f f T{A,B,A,B).
6.8.2 Let j/,^ be iT-categories: j/ is small and ^ is iT-complete.
Prove that the iT-category ir-Fun(j/, J^) of iT-functors and iT-natural
transformations is 1^-complete.
6.8.3 Let 1^ be a small symmetric monoidal category. Prove that 1^ is a
full subcategory of a symmetric monoidal closed category iV^^ the
structure of ir being induced by that of #^. [Hint: choose #^ = Fun(ir*, Set).]
6.8.4 Let j/ be a iT-category. Prove that a iT-functor F: s^ >ir is
iT-representable iflF limFlj/ exists and is preserved by F,
6.8.5 Consider R_|_ = {r| r G M, r > 0} U {+00} with the reversed poset
structure, i.e. there is a morphism r >s when r > s. Show that R_|_
is a complete and cocomplete symmetric monoidal closed category if we
define
r (g) 5 = r + 5, [5, <t] = max {t — 5,0};
as a matter of convention, 00 — 00 = 0.
6.8.6 If (X, d) is a metric space, prove that X together with the distance
d: X X X >R_|_ defines a category enriched in R_|_.
6.8.7 If (X, d) is a metric space viewed as a R_|_-category, prove that
a R_|_-functor (X, d) >R_|_ is just a mapping /: X >R such that
[/(x),/B/)] <d(x,2/), x,2/GX.
6.8.8 If (X, d) is a metric space and 1 is the singleton, show that a pair
of R_|_-adjoint R_|_-distributors /: 1 o > (X, d), g: (X, d) o >1, / H ^ is
just a pair of mappings /,^: X ]W , such that
B) Vx,2/GX /(x)+^B/)>d(x,2/).
6.8.9 Let (X, d) be a metric space and (X,d) its Cauchy completion,
obtained as the set of equivalence classes of Cauchy sequences in X.
Prove that the elements of X correspond bijectively with the pairs of
adjoint distributors as in 6.8.8. [Hint: given x G X, define /: X >R_|_
by f{x) = d{x,x) and put g = f] given an adjoint pair f,g choose

348 Enriched category theory
an ^ X such that /(a^) + gidn) < :^(^ ^ N*) ^nd show this yields a
Cauchy sequence.]
6.8.10 Conclude that a metric space (X, d) is complete in the classical
sense (i.e. every Cauchy sequence has a limit) iff every R_|_-distributor
1 o >{X,d) which has a right R_|_-adjoint is induced by an R_|_-functor
1 >(X,d); see 7.9.3, volume 1.

7
Topological categories
Categorical methods have proved to be particularly useful when studying
various questions in algebraic topology: today, most books on the subject
contain a crash course in category theory, which turns out to provide
a fruitful setting for handhng the required structures. In this topic, the
notions of abelian category and exact sequence play a key role.
This chapter is mainly concerned with the description of good
categorical settings for developing general topology. And if this deserves
a chapter in a book, this is clearly because the most obvious category
one could think of - the category of topological spaces and continuous
mappings - does not have rich categorical properties (like being
regular, monadic, cartesian closed, a topos, ...) which would have made
applicable the results of some other chapters of this book.
In topology, one is mainly concerned with the problem of convergence
and in particular problems of convergence in spaces of continuous
functions. One is particularly interested in situations where "if a sequence of
continuous functions converges, the limit is again continuous". In fact,
this requires a notion of convergence in the set of all functions (not
necessarily continuous)... to express finally the continuity of the limit. Such
a "good" situation is thus obtained when the set of continuous functions
is closed in the set of all functions, for the corresponding topology
inducing the notion of convergence. For example, in calculus, when studying
the functions from W^ to R'^, one puts a special emphasis on uniform
convergence on compact subsets ... and proves that if a sequence of
continuous functions converges in this sense, the limit is again continuous.
This is a striking difference from pointwise convergence, for which a limit
of continuous functions is in general not continuous. The previous
example uses explicitly the notion of "uniform structure", but topological
substitutes exist for defining, for continuous functions between topolog-
349

350 Topological categories
ical spaces, the notion of "uniform convergence on the compact subsets"
(via the compact-open topology) or the notion of pointwise convergence
(via the pointwise topology).
In categorical terms, one is thus interested in topologizing in a nice
way the spaces C{X,Y) of continuous functions, thus finally in
finding a good symmetric monoidal closed structure on the category Top
of topological spaces. A (unique) such structure exists: the
corresponding notion of convergence on the spaces C(X, Y) of continuous functions
is pointwise convergence... which, unfortunately, is not topologically
very interesting. Finally, the very bad thing is the fact that Top is not
cartesian closed: indeed, the categorical product in Top is the usual
topological product, thus a very good one from the point of view of topology;
therefore topologizing C(X, Y) in such a way to get a right adjoint to the
cartesian product would yield an ideal situation! But this is not always
possible. In fact it is possible when X is locally compact, in which case
the corresponding topology on C(X, Y) is the expected one, namely the
compact-open topology.
So Top is not cartesian closed; we have to live with this! Can one
find a good "cartesian closed approximation" of Top? For example by
dropping some "bad" topological spaces ... or by adding some new
objects or arrows to create the function spaces Y^ which do not exist! In
the Hausdorff case, we present an example of each type: the
subcategory of compactly generated spaces and the subcategory of compactly
continuous mappings ... and prove that these two categories are in fact
equivalent!
Finally we define those functors, called topological, which satisfy ax-
iomatically the basic properties of the forgetful functor Top >Set. The
key notion consists in axiomatizing categorically the notion of "initial
topology on a set for a given family of mappings to topological spaces".
7.1 Exponentiable spaces
As indicated in the introduction, one of the first questions asks if the
category Top of topological spaces and continuous mappings admits the
structure of a symmetric monoidal closed category (see chapter 6).
Proposition 7.1.1 Suppose the structure of a symmetric monoidal
closed category is given on Top. For two spaces X, Y:
A) the set underlying the tensor product X <SfY is necessarily the
cartesian product X xY of the underlying sets;

7,1 Exponentiable spaces 351
c(i, [1, X]) '^''^■'^•^" 1 c(i, [/,X])
'^"^'•^' c(H„i.i) '"'■"''•''''
Diagram 7.1
B) the set underlying the internal function space [X, Y] is necessarily
the set C(X, Y) of continuous mappings from X to Y;
C) the unit space I is necessarily the singleton.
Proof First of all, observe that the space / cannot be empty. Indeed
since 0 is the initial object of Top, it is preserved by every functor — 0
X, because such a functor admits [X, —] as a right adjoint (see 3.2.2,
volume 1). Therefore 7 = 0 would imply X^7(8)X^0(8)X^0for
every space X.
Since 7 is not empty, fix an arbitrary (obviously continuous) mapping
i: 1 >I from the singleton 1 to 7; considering the unique (obviously
continuous) mapping j: I >1, one has j oi = l-^ and thus 1 is
presented as a retract of 7. Every functor, thus in particular 1(8)—, preserves
retracts, thus 1 0 1 is a retract of 1 0 7 = 1. Therefore 1 0 1 is the
singleton or the empty set.
But 1 (8) 1 cannot be empty. Indeed, 1 0 1 = 0 would imply
C@,0)^CA01,0)^CA,[1,0]).
Thus l0 would correspond with a mapping k: 1 >[1,0] and the
composite koj: I >[1,0] with a continuous mapping 1 = 7(8) 1 >0 (see
6.1.7), which is a contradiction. So 1 (8) 1 is the singleton.
Thus Ij (8) j: 1 (8) 7 > 1 (8) 1 is a mapping from the singleton to itself,
so is the identity. The commutativity of diagram 7.1 indicates that the
upper horizontal arrow is a bijection, hence the continuous mapping
\j,lx]:[l,X] >[I,X]
is bijective for every X, Since this mapping has a continuous retraction,
namely [i, Ix], we know [j, Ix] and [i, Ix] are inverse homeomorphisms.
This implies immediately that i^j are themselves inverse
homeomorphisms: just put X = I and apply 6.1.7. So we have proved that 7 is
just the singleton.

352 Topological categories
Condition B) of the statement follows immediately. For two spaces
X, Y one has bijections
proving that the continuous mappings in C(X, Y) are indeed in bijection
with the elements of [X, Y].
It remains to prove condition A) of the statement. Consider again
two spaces X, Y and write Xd for the set X provided with the discrete
topology. Notice that in Top, Xd is just the copower Xd = Uxex-^*
The identity Ix- Xd >X is an epimorphism in Top. The functor
— <S)Y: Top >Top preserves colimits, because it has a right adjoint
[y, —]; see 3.2.2, volume 1. In particular it preserves coproducts and
epimorphisms (see 2.9.3, volume 1). One has therefore
^•^ ® ^ ^ (U.,;cO ^ ^ ^ II.,;.(i ^y)^ ]I.,^y ^^^-y-
This yields an epimorphism
XdxY^Xd^Y ^x^^y )X0y
On the other hand considering the morphisms a: X >1, /3: Y >1
one constructs
x®Y—^^^^ >X0i^x, x^y—^^^^ > 10y^y.
from which one gets a corresponding factorization
X 0 y >x X y
Let us prove that the composite
V a(8)ly /
V a(8)ly /
is the identity mapping on X x Y; this will prove that the surjection
Ix ® ly is also injective and thus bijective. Indeed the composite
XdxY^Xd^Y ^^®^y )X®y ^^^^ )X®i^x
has by definition the form
Uy >x (8) y—hL^A^x 01 ^ x.
x€X

7.1 Exponentiable spaces 353
Its composite with the injection sx of the coproduct is thus
or equivalently
Y^l®Y ^1^^ a^l ^^^1 .X^l^X
which is just the constant mapping on x since 1 0 1 = 1. An analogous
argument holds for the factor Y. D
So, by 7.1.1, a tensor product of two spaces X,Y must be the
cartesian product X X y of the underlying sets, provided with a convenient
topology. The most convenient one is clearly the product topology; being
allowed to choose it as a tensor product reduces to proving that Top is
cartesian closed (see 6.1.5), which unfortunately ... is false!
Proposition 7.1.2 The category of topological spaces is not cartesian
closed.
Proof Let us assume that the category Top of topological spaces and
continuous mappings is cartesian closed. We shall deduce a
contradiction. By 7.1.1 we know that for two topological spaces Y,Z, the
exponentiation Z^ is the set C{Y, Z) of continuous functions provided with
some topology. For this topology, the evaluation map
evyz: Z^ X Y >Z, {f,y) ^ f{y)
is continuous since it corresponds, by adjunction, with the identity on
Z^. Let us observe moreover that given a topology T on C(Y, Z) such
that the evaluation map
evyz: {C{Y,Z),T) x Y >Z
is continuous, T is finer than the topology of Z^: indeed, by adjunction,
the identity map C{Y, Z) = Z^ must be continuous. So the topology of
Z^ is the coarsest one making the evaluation map evy^ continuous.
In particular, let us choose for Z the unit interval [0,1] of the real Hne
and for Y the space Q of rational numbers, with the topology induced by
that of the real line M. The real line is a completely regular space, i.e. two
disjoint closed subsets can be separated by a continuous function to the
unit interval. This property is obviously inherited by every subspace, in
particular by Q. We have already proved that [0,1]*^ is the set C(Q, [0,1])

354 Topological categories
of continuous functions, provided with the coarsest topology making the
evaluation map
ev: C(Q, [0,1]) x Q .[0,1], (/, q) ^ f{q)
continuous. Let us deduce from this that Q is locally compact, which
will be a contradiction.
It is a matter of fact that Q is not locally compact. Observe indeed
that if r G M\Q is an irrational number, the closed interval [—r, +r] of
R determines by intersection with Q a closed neighbourhood of 0 in Q,
which we still write [—r, +r]. One has obviously [—r, +r] = [j^] — q, -\-q[
in Q, where q runs through the rational numbers 0 < q < r. Since
no finite subunion covers [—r,+r], [—r,+r] is not compact in Q. But
the local compactness of Q would imply the existence of a compact
neighbourhood V of 0 in Q. As a neighbourhood of 0, V contains a
neighbourhood of the form ] - q,-\-q[ with g G Q; choosing 0 < r < q
with r irrational, the closed subset [—r, +r] of Q is contained in the
compact subset V, thus is itself compact, which is a contradiction.
It remains thus to deduce the local compactness of Q from the
assumption of cartesian closedness of Top. We consider for this the continous
function So'- Q > [0,1] given by the constant mapping on 0. For every
rational number g G Q, 6o{q) G [0, |[ and [0, ^[ is open in [0,1]. By
continuity of the evaluation map, we can find Uq C [0,1]*^, Vq C Q, open
neighbourhoods of 6o and q such that ev(C/g xVq) C. [0, ^[. The closure Vq
of Vq is thus a neighbourhood of q and it suffices to prove it is compact.
Let us consider an open covering Ufc^K^^ ^^ ^- P^^^ii^g ^o = CK?
(the complement of the closure of V^), we get an open covering Uzgl^^
of Q, where L = K U {0}. We shall construct from this a topology
S on C(Q, [0,1]) making the evaluation map continuous. We take as
fundamental open subsets all the subsets
[C,J] = {g€C{q,[0,l])\9{C)Cj}
where C runs through the closed subsets of Q contained in some Wi,
/ G L, while J runs through the open subsets of [0,1]. To prove that S
makes the evaluation map continuous, consider g G C(Q, [0,1]), p G Q
and J an open neighbourhood of g{p) in [0,1]. p G W/q f^^ some lo ^ L
and g~^{J) is another open neighbourhood of p, since g is continuous.
By regularity of Q, there is an open neighbourhood W of p whose closure
is contained in Wi^ D g^^iJ)
peWcWcWi^ng-^{j),

7.1 Exponentiable spaces 355
This implies in particular g{W) C J, thus g G [W", J]. It remains to
observe that
ev([W,J] xW^CJ
which is just obvious.
Let us write T for the topology of [0,1]*^. It is the coarsest topology
making the evaluation map continuous. Since S is another topology of
this kind, S is finer than T. The T-neighbourhood Uq of 6o in C(Q, [0,1])
thus contains an 5-neighbourhood of Sq. This means that it is possible
to find indices /i,..., /n ^ L^ closed subsets Ci C Wi. in Q and open
subsets Ji in [0,1] such that
Soe[Ci,Wi,]n..,n[Cn,WiJcUq,
Let us prove that V^ C Ci U ... U C^.
If K ^ Ci U ... U Cn, fix a point p e Vq, p ^ CiU ., ,U Cn^ Then
{p} and Ci U ... U Cn are disjoint closed subsets in Q; by complete
regularity, we can find a continuous function /: Q >[0,1] such that
f{Ci U ... U Cn) = 0, f{p) = 1. In particular / coincides with Sq on
each Ci, thus 6o{Ci) = fiCi) C Wi.. This proves / G [Q, W/J for each
i = 1,... ,n, thus f E Uq. But this is impossible since by definition of
Uq,Vq,feUq and p eVq imply 0 < f{p) < ^. Therefore we must have
Vq C CiU .. .UCn- Since the Q's are closed, this yields
VqCVqCClU...UCnCWl,U,..UWl^.
Observe finally that if Wq = CVq appears in the right-hand term,
dropping it from the union will still produce a finite covering of Vq, proving
finally the compactness oiVq. D
Since the category of topological spaces is not cartesian closed, it is
sensible to make the following definition.
Definition 7.1.3 A topological space X is exponentiable when the
^^product functor^^
- X X: Top >Top
has a right adjoint.
We shall not give an explicit characterization of exponentiable spaces,
since the corresponding condition is rather technical. But we shall prove
an interesting sufficient condition for exponentiability.
Let us make clear that our topological spaces are not required to be
Hausdorff. In particular the notion of compactness does not require the
Hausdorff axiom. Now in the non-Hausdorff case, the notion of local

356 Topological categories
compactness should be made precise, since various definitions appear in
the literature, which are equivalent just in the Hausdorff case.
Definition 7.1.4 A topological space X is locally compact when every
neighbourhood of a point contains a compact neighbourhood of this
point.
Proposition 7.1.5 A locally compact space is exponentiable. The right
adjoint to the functor — xX is given by the functor mapping a space Y to
the set C(X, y) of continuous functions, topologized with the compact-
open topology.
Proof We recall that the compact open topology on C{X,Y) is that
determined by the fundamental open subsets
{K,U) = {f€C{X,Y)\f{K)CU}
where K runs through the compact subsets of X and U runs through the
open subsets of Y (our notion of compactness does not require Hausdorff-
ness). Let us write [X, Y] for the set C(X, Y) topologized in this way. It
is easy to observe that this construction is functorial in Y. Indeed given
a continuous mapping h: Y >Z, we must observe that composition
with /i,
[X,h]:[X,Y] >[X,Z],
is continuous. If AT C X is compact and V C Z is open,
[K,h]-\K,V) = {feC{X,Y)\ hf{K) C V}
= {feC{X,Y)\f{K)Ch-\V)}
^{K,h-\V))
is open since h is continuous.
Let us consider a continuous mapping g: Z x X >Y and the
corresponding function
r-Z >[X,Y], z^g{z,-).
To prove the continuity of 7, let us choose K compact in X and U
compact in F; then
7'\K,U) = {zeZ\\^xeK g{z,x) e U}, /
Fix a point z € 7~^(i<', C/); we shall construct an open neighbourhood W
of z contained in 7"^ (if, C/); this will prove that 7"^ (if, U) is open and
thus 7 is continuous. For every x £ K, g^^ (U) is an open neighbourhood

7.1 Exponentiable spaces 357
of (z, x), thus contains an open neighbourhood of the form Wx x T4.
The open subsets Vx cover the compact subset K, thus K is already
covered by finitely many of them, let us say 14^,..., Vx^. Putting W =
Wx-^ n ... n Wx^ we get an open neighbourhood of z. Choose z' G W.
For every x e K there is an index i such that x ^ Vx^; on the other
hand z' G M^i? thus g{z\x) G U. This proves that W is contained in
7-HK,U).
Conversely consider a continuous mapping /: Z >[X, y] and the
corresponding function
(p:ZxX >y, ^{z,x) = f{z){x).
To prove the continuity of (^, consider elements xEX^yEY^zEZ
such that ip{z^x) = y and an open neighbourhood U oiY; then
(^-i([/) = {{z',x') G Z X X| /(z')(^') ^ f^}-
Since /(z) is continuous, f[z)~^(U) is a neighbourhood of x. By
assumption on X, we can choose a compact neighbourhood K C f[z)~^(U) of
X. Since / is continuous
f-\K,U) = [z' G Z\\/x' G X /(z')(^') ^ U]
is open in Z. One obviously has
[z,x) e r^{K,U) X K Q^{U\
proving that (p~^{U) is a neighbourhood of {z,x). Thus (^ is continuous.
It is obvious that the previous data define natural inverse bijections.
D
Let us conclude this section with describing a symmetric monoidal
closed structure on Top. As proved in Pedicchio and Solimini, this is
in fact the unique symmetric monoidal closed structure existing on Top.
Proposition 7.1.6 There exists a symmetric monoidal closed structure
on Top for which the internal hom-functor [X, Y] is given by the space
C{X^Y) of continuous functions provided with the pointwise topology.
Proof We recall that the pointwise topology is obtained by choosing
as fundamental open subsets
{x,U) = {feC{X,Y)\fix)€U}
where x runs through the points of X and U runs through the open
subsets of Y. Let us write [X, Y] for the set C{X, Y) topologized in this
way. It is easy to observe that this construction is functorial in Y, Given

358 Topological categories
a continuous mapping h: Y >Z, we must prove that composition with
h is continuous:
[X,h]:[X,Y] >[X,Z].
Given x e K and V open in Z,
[X,h]-\x,V) = {fGC{X,Y)\ hfix) G V}
= {feC{X,Y)\f{x)eh-\V)}
= {x,h-\V))
is open since h is continuous.
To discover the form of the tensor product topology, consider a
continuous function /: Z >[-X', F] and the corresponding mapping
(fiZxx >y, (z,x)i-^/(z)(x),
where Z x X is the topological product of Z, X in Top. Since / is
continuous, fixing z ^ Z yields
(p{z,-):X >y, x^f{z){x),
which is the continuous function f{z). Fixing x E X yields
(p{-,x): Z >y, z^f{z){x).
Given an open subset U CY
<p{-,x)-\U)^{zeZ\f{z){x)eU}
^{z€Z\f{z)e{x,U)}
= r'{x,U),
which is open in Z, since / is continuous. Thus when / is continuous, if
is continuous in each variable.
Conversely consider a function (p: Z x X >Y which is continuous
in each variable. In particular, fixing z E Z, the function
(p{z, -): X eY, X H^ ip{z, x)
is continuous, yielding a mapping
f:Z >[X,Y], z^ifiz,-).
This mapping / is itself continuous since, given x E X and U open\n
Y,
f-^{x,U) = {z€Z\^iz,x)eU}
= ¥>(-, x)-i(C/),

1,2 Compactly generated spaces 359
Figure 7.2
which is open in Z since ip{—^x) is continuous.
So we have already constructed bijections
SC{ZxX,Y) ^C{Z,[X,Y])
where SC stands for "separately continuous", i.e. continuous in each
variable. These bijections are obviously natural and it remains to find
natural bijections
C{Z^X,Y)^SC{ZxX,Y).
We know that Z (S> X must be the product set of Z, X (see 7.1.1). Thus
the problem is to find a topology Z <S) X on the product set Z x X
such that being continuous on Z 0 X is equivalent to being separately
continuous on the topological product. Since a function /: Z x X >Y
is separately continuous precisely when the composites
zAzi^zxx—^—>y, X ^^^"^ )Zxx—^—>y
are continuous for all x G X, z G Z, the topology of Z 0 X is just the
final topology for all the mappings
zAzi^zxx, X ^^'~hzxX. n
The topology of Z 0 X is sometimes referred to as the topology of
"asterisks". In M^, figure 7.2 is an asterisk, where the boundary is not
in [/, except the central point: all "horizontal" and "vertical" sections
of this asterisk are open subsets of the corresponding real lines. This is
thus a neighbourhood of the central point for the topology of asterisks.
7.2 Compactly generated spaces
Proposition 7.1.5 could suggest one restricts one's attention to locally
compact spaces in order to find a cartesian closed subcategory of Top.

360 Topological categories
This would be very restrictive (a Banach space is locally compact when
it is finite dimensional!) and moreover would not work, since the space
C{X,Y) of continuous functions between locally compact spaces X, y,
provided with the compact open topology, is in general not locally
compact.
Let us recall that the crucial idea behind the compact open topology
is to describe a topology nicely related with the notion of "uniform
convergence on compact subsets". This suggests we weaken the notion
of continuity in the following way:
Definition 7.2.1 Let X,Y be topological spaces. An arbitrary
mapping f: X >Y is compactly continuous when its restriction to each
compact subset of X is continuous.
This notion of "compact continuity" will turn out to be stable under
a limit process which is uniform on the compact subsets.
Obviously, one has
Proposition 7.2.2
A) The composite of two compactly continuous functions is compactly
continuous.
B) Every continuous function is compactly continuous.
C) If X is a locally compact space, a function f: X >Y is compactly
continuous if and only if it is continuous.
Proof If /: X >y, g: Y >Z are compactly continuous, for every
compact subset K C X the continuity of /: K >Y implies that f{K)
is compact in Y. Since g is continuous on f{K) by assumption, f o g is
continuous on K.
The second statement is obvious. The third statement reduces to the
fact that a function is continuous as long as it is continuous on a
neighbourhood of every point. D
One of the essential features of the notion of continuity is its local
character: a function is continuous if and only if it is continuous in
the neighbourhood of every point. The notion of compact continuity
does not share this property ... except if we assume that each compact
subset is also locally compact. This property is automatic when the
space is HausdorflF. Moreover when studying problems of convergenc^on
compact subsets, it is convenient to assume that the compact subsets
are closed: again this is automatic for HausdorfF spaces. For this reason,
in this section, we shaU restrict our attention to the case of HausdorflF
spaces.

1,2 Compactly generated spaces 361
It is probably useful to underline the fact that for non-Hausdorff
spaces, it does not suffice to replace "compact" by "compact locally
compact", or "compact closed", or "compact closed locally compact" to
get a direct generalization of the following results. Indeed, while
compactness is preserved by continuous mappings - an important fact in
what we shall do - closedness and local compactness are not.
Proposition 7.2.3 The category CC-Covf\p of compact Hausdorff spaces
and compactly continuous mappings is cartesian closed.
Proof The singleton is obviously a terminal object in CC-Comp.
Moreover if X, Y are Hausdorff spaces, their topological product X x y is
also their product in CC-Comp. Indeed, given two compactly continuous
mappings /: Z >X, g: Z >Y^ the unique factorization
@
^X xY
in Set is compactly continuous, since given a compact subset AT C Z,
the universal property of the topological product X x Y reduces the
continuity of {^) on K to the continuity of both / and g on K,
Now given two Hausdorff spaces F, Z, let us write CC{Y, Z) for the
set of compactly continuous functions from Y to Z provided with the
topology whose fundamental open subsets are given by
[K,U] = {feCCiY,Z)\fiK)CU}
where K is compact in Y and U is open in Z (the "compact open
topology" for compactly continuous mappings).
First of all, we must prove that the topology on CC{Y, Z) is Hausdorff.
If /,p: Y ^Z are distinct compactly continuous mappings, choose an
element y eY such that f{y) ^ g{y). Since Z is Hausdorff, choose [/, V
disjoint open subsets such that f{y) G [/, g{y) G V, The singleton {y} is
compact in Y and one has / G [{y}, U\,g e [{y}, V]. Moreover [{y}, U\
and [{2/}, V^] are disjoint, just because U and V are disjoint.
Next, let us prove that the evaluation mapping
ev: COY, Z) x Y >Z, (/, y) ^ f{y)
is compactly continuous, i.e. is continuous on every compact subset P
of the product. Since the product is the topological one, the projections
map P to compact subsets C C CC{Y, Z) and K C Y, Since P CCxK,
it suffices to prove the continuity of the evaluation on C x K, To do
this choose {f^y) € C x K and an open neighbourhood U Q Z of f{y).

362 Topological categories
Since /: K >Z is continuous, there exists a neighbourhood V of y
in K such that f{V) C U, Since K is compact Hausdorff, it is locally
compact and we can choose a compact neighbourhood L C V oi y in
K\ in particular /(L) C U. Now [L, [/] D C is a neighbourhood of / in
C and L is a neighbourhood of y in AT. Therefore ([L, [/] D C) x L is a
neighbourhood of (/, y)\nCxK and, by definition, is mapped in U by
the evaluation mapping. This proves the continuity of the evaluation on
C X K and thus on P.
Finally consider a Hausdorff space X and a compactly continuous
mapping f: X x Y >Z, We must still prove the compact continuity
of the transposed mapping
g:X >CC{Y,Zl x^/(x,-),
that is, the unique mapping such that ev o (p x ly) = /. First of all
observe that the mapping g is well-defined, i.e. /(x, —) is compactly
continuous for each x e X. Indeed, this is just the composite of the
compactly continuous mapping / with the continuous mapping
Y >X X Y, y^{x,y).
To prove that g itself is compactly continuous, consider compact subsets
A Q X^ K C. Y and an open subset U C. Z. We must prove that g
restricted to A is continuous, i.e.
g-\[K,U])nA = {xeA\'iyeK f{x,y)eU]
is open in A, Fix x G p~^ ([AT, [/]). For every y ^ K^ the continuity of / on
the compact subset Ax K implies the existence of two neighbourhoods
T^ of X in A and Wy of y in AT, such that /(T^ x Wy) C [/. Since
K is compact, there exist finitely many elements 2/1,..., 2/n ^ K such
that Wy^,..., Wy^ already cover K, But V = V^^ D ... D Vy^ is still a
neighbourhood of x in A and
f(y XK)= iX^JiV X Wy,) C [X^JiVy, X WyJ Q [/.
This implies x G V C ^-^([ii', i7]), thus g~'^[[K, U]) is a neighbourhood
of each of its points x, D
Via proposition 7.2.3, we know how to enlarge the category Haus of
HausdorfF spaces and continuous mappings in order to get a cartesi^
closed category, with the exponentiation Z^ given by the set of arrows
from Y to Z, provided with the compact open topology. This is nice,
but the reader can argue that he does not want to weaken the notion

1,2 Compactly generated spaces 363
of continuity to that of compact continuity. Never mind, the category
CC-Comp is in fact equivalent to a full subcategory of the category Haus
of Hausdorff spaces and continuous mappings!
Proposition 7.2.4 The canonical inclusion Haus C CC-Comp admits a
left adjoint functor which is full and faithful. Thus CC-Comp is equivalent
to a full subcategory of Haus.
Proof Given a Hausdorff space (X, T), where T is the topology on the
set X, let us first construct another topology /C(T) on the same set X.
A subset [/ C X is /C(T)-open if and only if for every T-compact subset
K C X, KnU is T-open in K, Equivalently C C X is /C(r)-closed if
and only if for every T-compact subset K C X^ K DC is T-closed in
AT, i.e. T-closed in X since K itself is T-closed in X. Since intersecting
with K C. X distributes over arbitrary set-theoretical intersections and
unions, /C(T) is a topology on X. By construction of/C(T), every T-open
subset is /C(T)-open, thus /C(T) is finer than T. In particular /C(T) is a
Hausdorff topology and the identity mapping \x'- {X, ^(T)) > (X, T)
is continuous.
Next, let us prove that the two Hausdorff topologies T and /C(T)
have the same compact subsets and coincide on those compact subsets.
Since /C(T) is finer than T, every /C(T)-compact subset is certainly T-
compact. Conversely if K is T-compact and di^jCi C K, with each
Ci /C(T)-closed, we have in fact Clieii^i r\ K) C K with each CiOK
T-closed, by definition of/C(T). Thus a finite sub-intersection is already
contained in K, proving that K is /C(T)-compact. Since we know that
/C(T) is finer than T, it remains to prove that given K compact and
C C. K with C /C(T)-closed in AT, C is also T-closed in K. But since
K is /C(T)-closed, C is /C(T)-closed in X thus C = KnC is T-closed
in AT, by definition of /C(T). This proves that the identity mapping
Ix- (-X', T) >(X,/C(T)) is an isomorphism in CC-Comp.
Now we verify the functoriality of the construction mapping a
Hausdorff space (X, T) to the corresponding space (X, /C(T)). Given a
compactly continuous mapping /: (X, T) ^(^?«5) to another Hausdorff
space (y, 5), we must prove that /: (X, /C(T)) > (y, /CE)) is
continuous. Given C C y which is /CE)-closed, we must prove that f~^{C)
is /C(T)-closed, i.e. for every T-compact subset K Q X, f~^{C) f\K is
T-closed. But since K is T-compact, f{K) is 5-compact and
' f-^{C) nK = p{cn f{K)) n K,
Since f{K) is 5-compact and C is /CE)-closed, C n f{K) is 5-closed.

364 Topological categories
Thus f~^{C n f{K)) nK is T-closed in K, thus in X, by continuity of
fonK,
We have thus defined a functor /C: CC-Comp >Haus; on the other
hand we write t: Haus >CC-Con\p for the canonical (non-full)
inclusion. There are natural transformations £:: /Co/, =^ id, r/: \d => toJC given
by the identity mappings. The triangular identities of 3.1.5, volume 1,
are then clearly satisfied, from which adjunction follows. Finally, /C is
full and faithful since rj is an isomorphism, see 3.4.1, volume 1. D
CC-Comp is thus equivalent to the full subcategory of Haus generated
by those spaces (X, T) for which T = /C(T).
Definition 7.2.5 A Hausdorff topological space (X, T) is compactly
generated when a subset C C X is closed as long as its intersection with
every compact subset K C. X is closed. Such a space is also called a
"Kelley space".
Corollary 7.2.6 The category of compactly generated spaces and
continuous mappings is equivalent to the category of Hausdorff spaces and
compactly continuous mappings. It is a cartesian closed, full coreRective
subcategory of the category of Hausdorff spaces and continuous
mappings, n
Examples 7.2.7
7.2.7.SL Every locally compact Hausdorff space (X, T) is compactly
generated.
Indeed if [/ C X is /C(T)-open and x e U^ choose a compact T-
neighbourhood K of x. But KDU is T-open in K, thus a T-neighbour-
hood of X in K. Since K itself is a T-neighbourhood oi x in X, K OU
and thus U are T-neighbourhoods of x in X. So [/ is a T-neighbourhood
of each of its points, thus is T-open.
7.2.7.b A HausdorfF space which satisfies the first axiom of countability
is compactly generated.
So we assume that X is a HausdorflF space in which each point has
a denumerable basis of neighbourhoods. We consider a subset S C. X
which is not closed and construct a compact subset K C X such that
S n K is not closed.
Since S is not closed, we choose x G S\S. Consider a denumerable
basis {Vn)n£N of neighbourhoods of x and, for every integer n G N, a
point Xn G S nVn- Consider the subset K constituted of x and the
various Xn, n G N. Since every open covering {Ui)i^i of K is such that

1,2 Compactly generated spaces 365
X E UiQ^ for some index io we know K is compact. Since the Vn? ^ ^ N,
constitute a basis of neighbourhoods of x, there exists no such that
Kio ^ ^io' S^ ^io contains x and all the elements a:^, n > no, and it
remains to consider finitely many Ui containing the remaining elements
xo,..., Xn-i' Thus K is indeed compact and K OS is just the sequence
of elements {xn)n£N^ which is not closed since x is not in it {x ^ S) but
belongs to its closure (every neighbourhood of x contains some Vn which
contains x^).
7.2.7.C Every metric space is compactly generated.
A metric space is Hausdorff, since two points x, 2/ at a distance d are
contained in the open balls jB(x, |), B{y, |) which are disjoint. It satisfies
also the first axiom of countability because the open balls B{x^ ^), n G
N*, constitute a basis of neighbourhoods of x. One gets the conclusion
by 7.2.6.b.
Let us conclude this section with a warning. In the category CC-Comp
of Hausdorff spaces and compactly continuous mappings, the cartesian
closed structure is given by the topological product and the compact-
open topology. Going through the equivalence with the category C^-Haus
of compactly generated spaces, one gets the corresponding cartesian
closed structure on C^-Haus. The cartesian product in C^-Haus is a priori
no longer the usual topological product. Let us prove that it will be,
under an assumption of local compactness.
Lemma 7.2.8 For a Hausdorff space (X, T), the following conditions
are equivalent:
A) (X, T) is compactly generated;
B) T is the final topology for all the inclusions K ^^ X, where K runs
through all the compact subspaces of (X, T);
C) T is the final topology for a family {ff. Ki >X)i^i of mappings,
with each Ki a compact Hausdorff space;
D) T is the final topology for the class of all continuous mappings
f: K > (X, T), where K runs through all the compact Hausdorff
spaces.
Proof A) =^ B) is just the definition of a compactly generated
space. B) =^ C) is obvious. C) => D) is obvious since adding
already continuous functions to a family does not change the finality
of the original topology. To prove D) =^ A ), choose U C X whose
intersection with each compact subset C C X is open in C. For each
continuous mapping /: K >X with K compact, f{K) is a compact

366 Topological categories
subset of X and
f-\u)^r'{unf{K))
is therefore open in K. By the finality of the topology T, U is open in
X. D
Proposition 7.2.9 Let Y be a locally compact Hausdorff space. In the
cartesian closed category of compactly generated spaces, each cartesian
product X xY is the usual topological product.
Proof Let X, Z be topological spaces with X compactly generated. We
write X X y for the topological product and Z^ for the set of continuous
mappings provided with the compact-open topology. By 7.1.5 a mapping
/: X X Y >Z is continuous if and only if the corresponding mapping
/: X >Z^ is continuous. By 7.2.8, / is continuous when its restriction
to each compact subset K C X is continuous. Applying 7.1.5 again, we
find that / is continuous if and only if its restriction to each subspace
if X y is continuous, where K runs through the compact subsets of X.
Since each point of Y has a compact neighbourhood, the continuity of /
on X X y is still equivalent to its continuity on each subset KxK' ^ with
K C. X and K' QY compact subspaces. In other words the Hausdorff
space X X y has the final topology for all those inclusions K x K' C.
X X y. Since each K x K' is compact, one concludes by 7.2.8.C) that
X X y is compactly generated. D
7.3 Topological functors
One typical feature about topological spaces is the possibility of
defining initial or final topologies. Given a family (Xi,7i)i^/ of topological
spaces and given mappings fi: X >Xi from a set X, there exists a
"best" topology on X making all the fis continuous: this is the "initial
topology" for the fis (it is generated by all the fi{U), for alH G / and
U open in X^). In this example, we were deahng with a discrete
diagram (Xi, Ti)i^i of topological spaces. Here is the general notion for an
arbitrary diagram.
Definition 7.3.1 Let U: si >^ be a functor. Consider ,
A) a category ® and a functor H: S) >si,
B) a cone (/d: B ^UH{D))^^^ onUoH in ^,
An initial structure for those data is a cone {qd'- A ^H{^))d^q, ^^
H such that:

1.3 Topological functors 367
A) U{A) = B and, for all D e 2 , fo = U^go);
B) if {jiD'- A' >H{D))j^^^ is a cone on the functor H whose image
(Uiho)'' U{A') >UH{D))j^^ under U factors via a morphism
b: U{A') >U{A) through (/d: U{A) >UH{D))^^^, there
exists a unique morphism a: A! >A such that U{a) = b and ho =
go ^ ci for every D E S^.
As usual, the uniqueness of the arrow a implies the uniqueness up to
isomorphism of the initial structure.
Definition 7.3.2 Let U: sd > J^ he a functor.
A) Given a category Q), U has initial structures of shape Q) if for
every functor H: Q) > J^ and every cone on U o H, there exists a
corresponding initial structure.
B) The functor U has initial structures when it has initial structures of
shape Q) for every small category Q).
C) The functor U is topological when it has initial structures of shape
S) for every category Q).
Examples 7.3.3
7.3.3.a The forgetful functor [/: Top >Set is topological. With the
notation of 7.3.1, it suffices to define A as the set B provided with the
initial topology for all the mappings /d: the topology generated by all
the subsets /^H^)? with V open in H{D).
7.3.3.b If Cat is the category of small categories and functors, the
functor U: Cat >Set mapping a small category ^ to its set of objects
has initial structures. With the notation of 7.3.1, the set B is made
a category with S as a set of objects by choosing SF,6') to be the
projective limit
B{b,b') = \imH{D){fD{b)jD{b')).
7.3.3.C A functor H: Q) >sd has a limit precisely when there
exists an initial structure along the functor JJ\ sd >1 (where 1 is the
terminal category) for the unique possible cone on U o H.
The difference between topological functors and functors having initial
structures is not easy to grasp intuitively: it is just a question of size of
the diagrams involved in the problem. But we have already seen in 2.7.1,
volume 1, that requiring a property for all diagrams (possibly large) can
have drastic consequences. This is the case for topological functors:

368 Topological categories
Proposition 7.3.4 Every topological functor is faithful
Proof Consider a topological functor U: s^ >^ and two morphisms
u^v\ X > V in j/, such that U{u) = U{v). Consider the discrete
category ^ whose objects are all the arrows d of j/. Take H: Q) > j/ to be
the constant functor on Y and define a cone [jd'- B >UH{d))^^^ by
putting B = U{X) and fd = U{u). Write {gd: A >H{d))^^^ for the
corresponding initial structure. Define the cone (/i^: X ^■^i^))d£9
by
u if d E j/(X, A) and gdO d = v,
^ V otherwise.
Since U{u) = U(v), the cone {U{hd))^^^ coincides with the cone {fd)d£9,
from which there is a unique morphism a: X >A such that U{a) =
lu{X) and gd^ ci = hd. Putting d = a, we get in particular ga^ ct =
kali ha 7^ V, by definition of ha we get ha = f, thus a contradiction. Thus
ha = V and so by definition oi ha^ ha = u and finally u = v = ha. D
Since topological functors are faithful, the following lemma is often
useful:
Lemma 7.3.5 Let F: j/ ^^ be a faithful functor, H: Q) >j/ a
diagram in s^ and (/d: B >FH{D)) a cone on F o H. There
exists an initial structure for these data as long as such a structure
exists for the corresponding discrete diagram.
Proof Let us write V for the discrete category having the same
objects as ^. Consider an initial structure [go- A >H{D))j^^ for
the diagram H: V >j/ and the cone {fD)D€9' Foi" every morphism
d: D >D' of ^,
U{H{d) o go) = UH{d) o U{gD) = UH{d) o fj, = fj,, = U[gD').
from which H(d) o ^^ = g^f since U is faithful. This yields the result.
D
Proposition 7.3.6 Every topological functor is also cotopological.
Proof By "cotopological", we mean clearly the notion dual to "topo-
logical". I
A topological functor U: si >0i is faithful (see 7.3.4), thus by 7.3.5
it suffices to consider a discrete category ®, a functor H.\ 2 >j/ and
a cocone (/d: FH{D) ^^)De^' ^^^ ^^ consider the discrete category
^' whose objects are the pairs (X, 6), with X e s/ and b: B >U{X)

1.3 Topological functors 369
an arrow such that the composites bo fo have the form Uiao^h)', with
aD,h'' H{D) >X in j/, for each object D e S^. We define a functor
H'[ Q' >^ by putting H'(X, b) = X and a cone f[^^^y B >F{X)
onFoH' by putting /(x,6) ~ ^* -^^ assumption, we find an initial
structure fefe)- ^ ^^) (x,B)€^'* F™US D e Si, the cone (aD,6)(x,6)€^'
is such that U{aD,b) factors through fi-^ b) ~ ^ ^^^ f^] therefore we get
a unique go- H{D) >A such that U^go) = fo and g'^^ ^^ og^ = ao^b-
This yields the required cocone (^d: H[D) ^^)d^^ mapped by U to
{fo:FH{D)^B)j^^^.
To prove the universal property of {gD)D£9^ we must consider a cocone
[ho: H{D) ^^Od€^ ^^ '^ ^^^ ^ morphism 6: F(A) >F{A^) such
that boU{gD) = U{hD)' The pair (A', b) is an object of S', thus putting
a = g'rj^, ^N we get a morphism a: A >A' such that U{a) = b and
ao go = ciD.b' Prom the relations
U{a o gn) = U{a) o Uigo) = b o Uigo) = Uiho)
and the faithfulness of [/, we get a o go = hn- The uniqueness of a
follows again from the faithfulness of U and the relation U{a) = b. D
Proposition 7.3.7 A topological functor has both a right and a left
adjoint and these adjoint functors are full and faithful
Proof By 7.3.6, it suffices to prove the existence of a full and
faithful right adjoint to a topological functor U: si >^. Given an
object 5 G ^, we consider the empty diagram Q) (notation of 7.3.1) and
the corresponding initial structures for the empty cone of vertex B,
This is an object A ^ si such that f/(A) = B and for every A! ^ si
and b\ f/(A') >S, there exists a unique arrow a: A! >A such that
U{o) — b. This is precisely saying that the pair (A, 1b) is the coreflection
of B along U (see 3.1.1, volume 1). So U has a right adjoint functor V
(see 3.1.3, volume 1). Since each couniversal morphism UV{S) >B
is an isomorphism (the identity on S) the functor V is full and faithful
(see 3.4.1, volume 1). D
Following the classical terminology for the topological functor
C/:Top >Set, {X,T)^X
the right adjoint to a topological functor is called the "discrete object
functor" and the left adjoint is called the "indiscrete" or "chaotic object
functor".

370 Topological categories
Proposition 7.3.8 A topological functor creates limits and colimits.
Proof Consider a topological functor U: s/ >^ and an arbitrary
functor H: Q) >j/ such that the composite U oB. admits in J^ a limit
(pp: L >UH{D))^^^. The initial structure {go: A >H{D))^^^
for these data yields the limit oi H in j/. Indeed given another cone
(/i£): A' >H{D))j^^^ on the functor i?, the cone (^^(^d))^^^ factors
uniquely through the limit cone {pd)d£^ via an arrow b: U{A') >L;
this yields a factorization a: A! >L such that go ^ ci = ho and
U{a) = b. The uniqueness of a follows immediately from that of b and
the faithfulness of U. The case of colimits follows by duality (see 7.3.6).
D
When U: si >^ is a topological functor, all the objects A G j/
mapped to the same object B £ ^ can be thought of as "all the U-
topological structures which can be put on the object S". As for ordinary
topologies, one has the following result:
Proposition 7.3.9 Let U: si >^ be a faithful functor with initial
structures. For an object B e ^, the objects A e s/ such that U{A) = B
and morphisms f E s/ such that U{f) = 1b constitute a complete
preordered class. When U is a topological functor, all limits (even large)
exist in this preordered class.
Proof If U{A) = B = U{A'), by faithfulness of U there is at most
one morphism A >A' mapped to 1b; so we indeed get a preordered
class. Now given a family {Ai)i^i of objects mapped to S, the identity
morphisms Ab: B >U{Ai))^^j constitute a cone for which there
exists a corresponding initial structure {gii A >Ai)i^i. The universal
property of this last cone means precisely that A is the infimum of the
family {Ai)i^j in the preordered class indicated.
When U is topological, the assumption that / is a set can be dropped
(see 7.3.2). D
In the basic case of the topological functor U: Top >Set, on a given
set X there is just a set of possible topologies. So the preordered classes
of 7.3.9 are in fact complete preordered sets.
Definition 7.3.10 A functor U: s/ >^ is fibre small when, for every
object B G ^, the category of those objects A e ^ such that U{A) = B
and arrows f e s/ such that U{f) = 1b is equivalent to a small category.
The notion of "fibre smallness" is an alternative to the use of large
diagrams in the definition of a topological functor (see 7.3.2).

1,^. Exercises 371
Proposition 7.3.11 Let U\ si >^ be a functor. The following
conditions are equivalent:
A) U is topological and fibre small;
B) U has initial structures and is faithful and fibre small.
Proof By 7.3.4, one has A) =4> B). Conversely, by 7.3.5 it remains
to consider a discrete category ^, a functor H: Q) >j/ and a cone
(/d: B >FH{D))j^^^ on F o H and prove the existence of a
corresponding initial structure. For every object D G ^, let us consider
the one element cone fn- B >FH{D) which induces an initial lifting
vD' Ad >H{D), All the objects Ad are in the "fibre" over S, i.e.
U[Ad) = B\ since this "fibre" is equivalent to a small preordered set
(by assumption) there is just a set of isomorphism classes of such A/)'s;
since moreover this "fibre" is complete (by 7.3.9), we can choose A to
be the infimum of all the A/)'s. For every D G ^, we define qd to be the
composite
A "^ )Az, ^^ )jy(£>)
where ud expresses the relation A < Ad in the fibre over B,
Let us prove that [qd'- A >H{D))j^^^ is the initial structure
associated with the original data. First of all
U{gD) = U{vd) o U{ud) = fDolB = fn^
Next choose a cone (/id*. A' >H{D)) j^^^ in s/ and, in J^, a
factorization b: U{A^) >B such that fD^b = U{hD)' By
universality of vd^ there exists a unique morphism qd' A! >Ad such that
U{aD) = b and vd ^ cld = hD- But the cone [ud'- A >Ad)d£9 is
the initial lifting of the cone (a^: A' >Ad)d£Q} and the
factorization b: U[A') >B yields a unique morphism a: A' >A such that
U[a) = b and ud^ ci = cld- This implies
qd o a = vd o Ud o a = vd o cld = ^d
from which the result follows immediately. D
7.4 Exercises
7.4.1 Let F: Top >Top be a colimit preserving functor. Prove that at
the level of underlying sets, F is just the product with a fixed topological
space X,
7.4.2 Let F: Top >Top be a colimit preserving functor. Prove the
existence of a unique natural transformation from F to the identity.

372 Topological categories
7.4.3 Prove that the category of compactly generated spaces is complete
and cocomplete.
7.4.4 Prove that every closed subspace of a compactly generated space
is itself compactly generated.
7.4.5 Prove that every open subspace of a compactly generated space
is itself compactly generated.
7.4.6 Show that the product of uncountably many copies of the real
line is not a compactly generated space. [Hint: consider those families
{ri)i£i for which there exists n G N such that for all i, ri = n or r^ = 0,
the number of those components for which r^ = 0 being less that n.]
7.4.7 An uncountable power of the unit interval [0,1] is compact by
the Tychonoff theorem, thus a compactly generated space. Show that
the subspace of those families {ri)i^i for which r^ is never 0 or 1 is not
compactly generated, proving that a subspace of a compactly generated
space is in general not compactly generated. [Hint: use 7.4.6.]
7.4.8 Let Z be a To, Ti, T2, regular or completely regular space. Given a
space y, consider the set C{Y^ Z) of continuous functions provided with
the compact open topology. Prove that C{Y^ Z) is again, respectively, a
ro,ri,r2, regular or completely regular space.
7.4.9 Prove that a directed set of continuous functions from a compact
space A to a metric space B converges for the compact open topology
on C(A, B) if and only if it converges uniformly on A,
7.4.10 Prove that a directed set of continuous functions on a compactly
generated space A to a uniform space B converges for the compact open
topology on C(A, B) if and only if it converges uniformly on the compact
subsets of A,
7.4.11 Let Y, Z be topological spaces. If the corresponding evaluation
map C(y, Z) X Y >Z is continuous, when Ciy.Z) is provided with
the compact open topology, prove for every topological space X the
existence of natural bijections
Top(xxy,z) ^ Top(x, c(y,z)).
7.4.12 Let U: si >Set be a topological functor. Prove that in si every
arrow factors as a regular epimorphism followed by a monomorphism.
7.4.13 Let JJ\ si ^►Set be a topological functor. Prove that si has a
generator and a cogenerator.
7.4.14 Let [/: si ^►Set be a topological functor. Prove that si is
well-powered (resp., co-well-powered) if and only if U is fibre small.

8
Fibred categories
This chapter introduces the notion of fibration, which is a powerful tool
in category theory, but also presents a deep reflexion on the bases of
category theory themselves.
Very often in the previous chapters we have considered families of
objects or morphisms in a category ^. This suggests the study of the
category Set(^) of families of objects and morphisms in ^:
• the objects are the families {Ci)i^i where / is a set and Ci is an
object of ^;
• the morphisms /: {Ci)i^i ^{Dj)j^j are the pairs (a, {fi)i£i) with
a\ I >J a mapping in Set and each /^: Ci >Do,(^i) a morphism
in^.
Of special interest are the families {Ci)i^i and (l/, (/^i Ci >Di)i^j)
for a given set /, which constitute what will be called the "fibre" at /.
The obvious projection functor Set(^) >Set will be the basic example
of a fibration.
Working with set-indexed families as in the previous example very
often leads to constructions or arguments which look innocent... but are
formally incorrect! For example given an /-indexed family of morphisms
(/i* Ci >Di)i^j in Set(^), few people really worry about considering
the "set"
Iq = {i ^ I\ fi is a monomorphism}.
But the sentence "/^ is a monomorphism" means
VCg^ V^,7;G^(C,Ci) fiOu = fiOv=^u = v
... and no axiom of set theory will ever imply that /q is a set, since the
formula contains a quantifier acting on a variable C which runs through
373

374 Fibred categories
something (namely, ^) which is not a set! Well, most of us will answer:
"O.K., you are right, but who really cares? Leave this to the logicians,
we are doing mathematics." And to some extent such an attitude is not
really to blame: we "feel" that it should be possible to make things all
right. For example by using the axiom of universes (see 1.1.4, volume 1),
so that ^ becomes a set in a bigger universe and as a consequence /q
does as well; and since /q Q !•> lo is a set in the same universe as /.
Now suppose - and this is the essence of the theory of fibrations -
that we want to index things not just by the category of sets, but by
an arbitrary category S'. Thus we consider a "fibration" F: ^ ^i
and think of the objects and arrows of ^ as families indexed by the
objects of i. For example consider I £ S and X^Y ^ ^ such that
F[X) = I = F{Y) together with a morphism /: X >Y in J^ such
that F{f) = 1/. Even supposing that S' is complete, cocomplete or
whatever you want, you cannot possibly find a general construction in S'
producing a subobject Iq Q I which represents intuitively "those indices
for which / is a monomorphism", as in the set-indexed case. Even if
vaguely overlooking some logical problems makes you feel free to consider
that any formula (pona. given set / should define a subset /q of /, you will
certainly not pretend that such a careless attitude can be carried over to
a more "structured" category S' (like groups, topological spaces, ...) in
order to define a subobject Iq C I in S\ for example, the monomorphisms
between two abelian groups certainly do not constitute a subgroup of
the group of all homomorphisms! You will immediately point out that
the formula (^ should be somehow compatible with the "structure" of S
in order to define a subobject in S\ this is the very fundamental notion of
"definability" with respect to a fibration. And it is careless but common
to overlook the fact that in the case of sets the property of being a set
is not a structure at all.
8.1 Fibrations
We start with some definitions, fixing the terminology and the notation:
Definition 8.1.1 Let F\ ^ >S be a functor. Given an object I e S',
the fibre of F at I is the subcategory ^j of ^ defined in the following
way:
• an object X e ^ is in ^i when F{X) = /;
• if X,Y are objects in ^i, a morphism f: X >Y of ^ is in ^i
when F{f) = 1/.

8.1 Fibrations 375
q
r "^ .
FiZ) ^ )F(y)—^-^F(X)
Diagram 8.1
Obviously, the fibres of F in 8.1.1 are by no means full subcategories
of J^.
Definition 8.1.2 Let F: ^ >S be a functor and a: J >I a mor-
phism of S. An arrow f: Y >X of ^ is cartesian over a if:
A) F{f) = a;
B) given g: Z >X is a morphism of ^ such that F{g) factors as
ao 13, there exists a unique morphism h: Z >Y in J^ such that
F{h) = P and g = f oh (see diagram 8.1).
Definition 8.1.3 A functor F: J^ >S' is a fibration when for every
arrow a: J >I in i and every object X in the fibre over I, there
exists in ^ a cartesian morphism f: Y >X over a. We also call ^ a
''category fibred over S"\
There is a similar and equivalent way of presenting the definition of
a fibred category: be careful, do not read it (see 8.1.7) without reading
also the counterexample 8.1.8! There is also the notion of "indexed
category" as in Pare and Schumacher: be careful again, it is not formally
equivalent to the notion of fibred category; roughly speaking, it is a
fibred category in which a distinguished choice of cartesian morphisms
and coherent isomorphisms has been made once for all: it is something
intermediate between a fibration (as in 8.1.3) and a split fibration (as in
8.3.3).
Lemma 8.1.4 Let F: ^ >S be a fibration.
A) The composite of two cartesian morphisms is again a cartesian
morphism.
B) If f: Y >X and g: Z >X are two cartesian morphisms over
the same arrow a: J >I of S, there exists a unique isomorphism
h in the fibre ^j such that g = f o h.

376 Fibred categories
Proof Assume that /: Y >X is cartesian over a: J >I while
g: Z >Y is cartesian over C: K > J; choose morphisms h: V >X
and 7: F{V) >K such that a0/307 = F{h). Since / is cartesian over
a we get a unique k: V >Y such that F{k) = /Soj and fok = h. Since
g is cartesian over /3, we get a unique /: V >Z such that F{1) = 7 and
gol = k. This implies fogol = fok = h. The uniqueness condition is
obvious. Statement 2 is the usual uniqueness up to an isomorphism for
the solution of a universal problem. D
Definition 8.1.5 Let F: ^ ><f be a functor and a: J >I a
morphism of S. An arrow f: Y >X of ^ is precartesian over a if:
A) F{f) = a;
B) if g: Z >X is a morphism of ^ such that F{g) = a, there exists
a unique morphism h: Z >Y in the fibre J^j such that g = f oh.
Lemma 8.1.6 Let F: ^ ><f he a fibration. Consider a morphism
f: Y >X in i mapped to a: J >I hy F. The following conditions
are equivalent:
A) f is cartesian over a;
B) f is precartesian over a.
Proof Clearly, cartesianness implies precartesianness (make /3 =
Ij in definition 8.1.2). Conversely if /: Y >X is precartesian over
a: J >/, we can consider g: Z >X as cartesian over a, since F
is a fibration. Then /, g are both precartesian over a and therefore by
a classical argument for universal problems, isomorphic via an
isomorphism in the fibre over J. Since g is cartesian, / is cartesian as well.
D
Proposition 8.1.7 Let F: ^ >S be a functor. The following
conditions are equivalent.
A) F is a fibration;
B) (a) for every arrow a: J >I in i and every object X in the fibre
over IJ there exists in ^ a precartesian morphism f: Y >X
over a;
(b) the composite of two precartesian morphisms is again
precartesian.
Proof A) implies B) by lemmas 8.1.4 and 8.1.6. Conversely, let us
prove that every precartesian morphism /: Y >X over a: J >I
is also cartesian. Consider therefore a morphism g: Z >X such that

8.1 Fibrations
377
K
C
-^ J
a
7
Diagram 8.2
7 = F{g) factors as 7 = a o /3 in (f, with /3: K > J. Consider a
precartesian morphism h': Z' >Y over /3. By assumption the composite
joh!\ Z' >X is precartesian over ao/3 = 7, thus there exists a unique
morphism h!'\ Z >Z' in the fibre over K such that f oh' oh" = g.
Putting h = h'oh" yields F{h) = /3olj^ = /3 and foh = g,To prove the
uniqueness of /i, consider /i: Z >Y such that F{h) = f3 and f oh = g.
Since /i' is precartesian, there is a unique /i''' in the fibre over K such
that h' o h"' = 7i; from / o /i' o /i''' = f oh = g we deduce /i''' = /i'' and
thus h = h. n
► (f satisfying condition B) (a)
Counterexample 8.1.8
Here is an example of a functor F: ^ -
of 8.1.7 and which is not a fibration.
Just consider the situation of diagram 8.2 where a, /3,7 are the non-
obvious morphisms of (f, /, ^, /i, fc, / are the non-obvious morphisms of ^,
F(/) = a, F{g) = 7, F(h) - /3, F(fc) = 7, F(/) = 1^ and all diagrams
are commutative. It is obvious that in J^, the following morphisms are
precartesian: /, /i, fc and of course, all identity morphisms. So condition
B) (a) is satisfied. But g = f o h is not precartesian, since k does not
factor through g. And / is not cartesian, since F{k) = 7 factors through
F{f) = a, while k does not factor through /. So there is no cartesian
morphism over a.
Examples 8.1.9
8.1.9.a For every category ^, the identity functor on ^ is obviously a
fibration: every morphism is cartesian.

378 Fibred categories
8.1.9.b Given a category ^, consider the example (p: Set(^) >Set
described in the introduction to this chapter:
• the objects of Set(^) are the families {Ci)i^i where / is a set and
d is an object of ^;
• a morphism /: (G^)^^/ >{Dj)j^j in Set(^) is a pair {a,{fi)ia)
where
a: I >J is a mapping in Set and /^: Ci >Dc,(^i) is a morphism
in^;
• the composite
is the pair (/3 o a, (^f^co o fi)i£i)''>
• (^ maps (Ci)ie/ to / and (a, [fi)i£i) to a;
it is obvious that Set(^) is a category and (^ is a functor. Now given a
mapping a: J >I in Set and a family {Ci)i^i of objects of ^, define
£)^. = Coc{j) and /j! Dj ^^a{j) the identity morphism, for every index
j G J. This defines a morphism (a, {fj)j£j)' {Dj)j^j >{Ci)i^i which
is cartesian over a.
8.1.9.C Given a category ^ with finite limits, consider the category
Ar(^) of arrows of ^ as in 1.2.7.C, volume 1:
• the objects of Ar(^) are the triples (A,/, jB) where /: A >B is
an arrow of ^;
• a morphism (li, f): (A^f^B) >{C^g^D) is a pair of morphisms
u: A >C, v: B >D such that g ou = v o f.
The "codomain" functor
9i:Ar(^) >^, 9i(A,/,B) = B, 9i(^,^) = ^
is a fibration. Indeed, given a morphism f: jB >D in ^ and an object
(C,p, D) in Ar(^), computing the pullback in diagram 8.3 yields a
morphism {u,v): {A,f,B) >(C,g,D) in Ar(^) which, just by definition
of a pullback, is cartesian over v. In this example, observe that the fibre
over an object jB G ^ is exactly the category ^/jB, as defined in 1.2.7.a,
volume 1 - i.e. intuitively the category of jB-indexed families in ^. This
fibration is referred to as the canonical fibration of ^ over itself.
S.l.Q.d Write 1 for the terminal category, with a single object and just
the identity on it. For every category ^, the unique functor ^ >1 is

8.1 Fibrations
379
/
B
-> C
-> D
Diagram 8.3
a fibration since, given an object C G ^, the identity on C is obviously
a cartesian morphism.
Our next example is worth being singled out.
Example 8.1.10: The small fibrations.
Let us go back once more to the example of the introduction, also
described in S.l.Q.b, supposing now that the category ^ is small. This
ordinary small category ^ is thus a category internal to the category of
sets (see 8.1.1, volume 1) and we shall write it as ^ = (Co, Ci, do? di^i^ c),
where Co is the set of objects and Ci the set of morphisms. Given a set
/, an /-family [Mi)i^i of objects of ^ is just a mapping m: / >Co.
Consider another set J and a mapping n: J >Co corresponding to the
choice of a family {Nj)j^j of objects. Given a mapping a: I > J, a
family (/j: Mi ^^ot{i))i^j of morphisms of ^ corresponds to a
mapping /: / >Ci such that do o f = m^ di o f = n o a.
This construction generalizes immediately to the case of an internal
category ^ = (Co, Ci, do. di^i^ c) in a category S' with finite limits. One
defines a fibration ^: S(^) >S in the following way:
• the objects of S{^) are the pairs (/, m) where / is an object of S
and m: / >Co is a morphism of S\
• a morphism (/, m) > (J, n) of S[^) is a pair (a, /) with a: I >J
and /: / >Ci morphisms of (f satisfying the conditions doof = m^
dio f = noa;
• the composite of two morphisms
(j,^)_(^i/U(J,n)-^M_
■iK,l)
is the pair (/3 o a,m o /i), where /i: / >CiXcoCi is the unique
factorization through the pullback obtained from the relation diof =
no a = do o g o a;
(p{I, m) = I and (p(a, f) = a.

380 Fibred categories
It is routine to check that <f (^) is indeed a category and (^ is a functor.
Now given a morphism a: I >J in S' and an object (J, n) of <f (^),
one immediately gets a morphism
(a, z o n o a): (/, n o a) > (J, n)
of ^(^). It is cartesian since given a morphism G,/): {K,l) >{J,n)
such that 7 = a o /? for some /3: AT >/, the unique required
factorization {K,l) > (/, n o a) is the pair (/3, /).
Anticipating the considerations of the next section, we immediately
make the following definition:
Definition 8.1.11 A fibration F: ^ >S' over a category i with finite
limits is called a small fibration when it is equivalent, in the 2-category
Fib(^) offibrations over £, to the category (p: S'{^) ><f of 8.1.10, for
some internal category ^ of S.
Let us conclude with some general processes for building new fibra-
tions from given ones.
Proposition 8.1.12 IfF: ^ >S and G: S >^ are fibrations, then
so is the composite G o F: ^ >^.
Proof Consider a morphism a: A >B in ^ and an object X G ^
such that GF{X) = B. Since G is a fibration, there is a cartesian
morphism a: I >F{X) over a; since F is a fibration, there is a cartesian
morphism /: Y >X over a. It is then obvious that / is also cartesian
over a. D
Proposition 8.1.13 Given two categories #', S', the projection functor
p: ^ X S >S is a fibration.
Proof Given a morphism a: J >I in S' and an object (X,/) of
3F X S., the morphism (lx,Q^): (X, J) >(X,/) is obviously cartesian
over a. D
Proposition 8.1.14 If F: ^ ><^ and G: Q) >^ are hbrations,
then so is the functor F x G: ^ x Q) >S x ^.
I
Proof Given a morphism (a,a): (J,B) >{I^A.) \n S x^ and ani
object (X,M) m ^ xS), choose /: Y >X in S cartesian over a and
m: N >M in S) cartesian over a; it is obvious that the morphism
(/, m): (Y, N) >(X, M) is cartesian over (a, a). D

8.1 Fibrations 381
^ ——> ^
G
H
Diagram 8.4
Proposition 8.1.15 Consider a puUback of categories and functors as
in diagram 8.4. If F is a fibration, then so is G.
Proof Consider a morphism a: D >C in ^ and an object of ®
in the fibre over C, thus a pair (X, C) with X ^ ^ and H(C) =
F[X). Choose a cartesian morphism /: Y >X over the morphism
H{a): H{D) >H{C) = F{X) of ^. In particular the identities
F{Y) = h[d) and F(/) - H{a) hold, thus (/,a): (Y,D) >(X,C)
is a morphism of ®. It is obviously cartesian over a. D
Proposition 8.1.16 Let F: ^ >(f be a fibration and 3) a small
category. The functor of composition with F,
F^:^^ ,^^,
is also a fibration.
Proof Consider a functor M: Q) >S and a functor P\ 3) >^ in
the fibre over M; this means that F o P — M. Given another functor
N: 3) >i and a natural transformation /x: N =4> M, we must
construct a cartesian morphism r\\ Q ^ P over /x.
For each object £) G ®, choose over /xd- ^(^) >M{D) a cartesian
morphism tjd- Q{D) ^^(^)- Given an arrow d: D' >D in ®, the
naturality of /x and the cartesianness of r]D imply that P{d)or]D' factors
uniquely through r]D via a morphism Q(d): Q[D') >Q{D). This
defines the functor Q and the natural transformation 77, which is cartesian
over /x since each 77D is cartesian over fiD- CI
Corollary 8.1.17 Let F: ^ >i be a fibration and 3) a small
category. The puUback of diagram 8.5, where A(E) is the constant functor
onE eS, determines a fibration ^^^^ >S.
Proof Use 8.1.16 and 8.1.15. D
Observe that the fibre of J^(^) over an object I e ^ has for objects
those functors M: 3 >^ such that F o M is the constant functor on

382 Fibred categories
^{9) > ^9
A
Diagram 8.5
/, thus exactly the functors M: Q) > J^/. The fibration J^(^) can thus
be thought of as the fibration of diagrams of shape Q) in the fibres of
F: J^ ><f.
To avoid any ambiguity, let us make clear that the dual notion of a
cofibration is obtained by dualizing both categories J^ and i in definition
8.1.3:
Definition 8.1.18 A functor F: ^ ><f is a cofibration when the dual
functor F*: J^* >S'* is a fibration.
Notice that the notions of fibration or cofibration can also be
"dualized" to the case of contravariant functors, by dualizing just one of the
categories ^ or (f in definition 8.1.3.
8.2 Cartesian functors
Cartesian functors are the "morphisms of fibred categories".
Definition 8.2.1 Let F: ^ >S and G: ^ >S' be two fibrations
over the same base category S, A cartesian functor H: (J^, F) > (^, G)
is a functor H: ^ >^ such that:
A) GoH = F;
B) G maps a cartesian morphism for F to a cartesian morphism for G.
Definition 8.2.2 Let F: ^ xf and G: ^ ><f be two fibrations
over the same base category 8, IfH, K: (J^, F) ^ (^, G) are cartesian
functorSj a cartesian natural transformation 6: H => K is a natural
transformation such that G*9 = F,
It is obvious that the fibrations over a base category (f, together with
the cartesian functors and cartesian natural transformations between
them, constitute a 2-category written Fib((^). We often write Cart(^, ^)
for the category of cartesian functors and cartesian natural
transformations from (J^,F) to (^,G).

8.2 Cartesian functors 383
Examples 8.2.3
8.2.3.a Consider a functor F: ^ >S' between two categories and
the corresponding fibrations (p: Set(^) >Set, xp: Set(^) >Set as in
8.1.9.b. The functor F induces a cartesian functor $: Set(^) >Set(^)
over Set by putting
$((a)ie/) - iF{Ci)),^j, $(«,(/0i€/) = (a, (n/i)),e/).
where the notation is that of 8.1.9.b.
8.2.3.b The previous example can easily be generalized to the
context of 8.1.10. Consider a category S' with finite limits, two internal
categories ^,® and an internal functor F: ^ >®, given by the two
morphisms Fq: Cq >Do, Fi: Ci >Di. One immediately gets a
cartesian functor (f(F): (f(^) ^8{Qi) by putting, with the notation of
8.1.10, (r(F)(/,m) = (/,Foom) and (r(F)(a,/) = (a,Fio/).
8.2.3.C In example 8.2.3.b, if F, G\ ^ ^Q) are two internal functors
and 0: F =4> G is an internal natural transformation, given thus by
a morphism d\ ^o ^^i, one gets a corresponding cartesian natural
transformation S{d)\ <f(F) ^ g{G) by putting (^@)(/,m) = (l/,6>om).
Proposition 8.2.4 Consider a category § with Gnite limits and two
internal categories ^,®. The constructions of 8.1.10 and 8.2,3.b,c yield
a full and faithful functor
lnt(^, &) > Cart((r(^), (r(®))
where lnt(^,®) denotes the category of internal functors and internal
natural transformations from ^ to S>,
Proof Checking the functoriahty is just routine. The faithfulness is
easy. Indeed given internal functors F, G: ^ ^& and internal natural
transformations O^t. F ^ G^ one has by definition
from which the faithfulness follows.
The fullness of the construction results from the same relations. If
6: S{F) => S{G) is a cartesian natural transformation, then for each
object (/,m) G S{^) the relation S{3}) * 6 = S{^) implies that 6G,^)
is in the fibre over /. Therefore one has
©(/,m) = A/, /): {I, Fo o m) > (/, Go o m)
with dQof = Foom and diof = G^om. In particular 6(Co,ico) = iXco.O)
for some 9: Cq >Di satisfying do o 9 = Fq, di o 0 = Gq. We leave to

384 Fibred categories
the reader the straightforward verification that the usual naturality of
O implies the condition on the morphism 9 that it induces an internal
natural transformation 6: F =^ G. D
Proposition 8.2.4 can be interpreted as the fact that the 2-category of
internal categories in S' is (equivalent to) a full sub-2-category of Fib(^).
On the other hand S' itself is a full subcategory of the category of internal
categories in S':
• each object I e S' determines the "discrete" internal category dl on
/, namely dl = (/, /, 1/, 1/, 1/, 1/) (see 8.1.6.a, volume 1);
• each morphism a: I > J in S' determines a corresponding internal
functor da: dl >dJ just via the pair da = (a, a).
It is obvious from 8.1.2, volume 1, that this construction yields a full and
faithful functor, because the axioms of internal functor F: dl >dJ
reduce immediately to the single relation Fq = Fi. Composing this
embedding with the functors of 8.2.4 thus yields a functor
^ >Fib(^), I^S{dI), a^S{da).
Observe that since we are just working with discrete internal categories,
the existence of finite limits in S is not necessary.
It is probably useful to compute explicitly the values of S{dl) and
S{da).
Lemma 8.2.5 Let I ^ S be an object of an arbitrary category. With the
previous notation, the fibration S{dl) over S is just the source functor
S/I >S, {K,E)^K.
If a: I >J is a morphism of S, the corresponding cartesian functor
S/I >S/J, {K,C) >{K,aop)
is just the composition with a.
Proof By 8.1.10, an object of S'{dl) is indeed a pair {K,C) where
/3: K >I is a morphism of S'. A morphism {K, /3) > (L, 7) in ^(9/)
is then a pair {e^ip) where e: K >L and (p: K >/ are morphisms
of S satisfying /3 = (^, 7 o g: = (^; this is indeed equivalent to giving
e: K >L such that 7 o g: = /3.
Now given a: / >J, S{doL) maps the object {K,E) € ^/I to the
pair (jRT, a o /3) and the morphism (g:, (p): (K, C) > (L, 7) on (g:, a o (^);
this is again composition with a. D

8.2 Cartesian functors 385
Proposition 8.2.6 For an arbitrary category S^ the functor
S >Fib(^), I^S[dI), a^S^da)
is full and faithful
Proof Given a morphism a: I >J in ^, S'{da) is just the composition
with a (see 8.2.5) so that ^(9q;)(/, 1/) = (/, a). This proves faithfulness.
Fullness follows from the same formula. If H: S/I >S/J is a
cartesian functor, from i/(/, 1/) = (/, a) we get a morphism a: / >J.
Since H is required to commute with the source functors, a morphism
/3: (K,fc) >(L,/) in S/I is mapped by H onto /3: {K,k') >{L,V)
in SI J. In particular I o p = k and V o ^ = fc^ Considering a morphism
k\ {K,k) >(/, 1/) yields fc' = aofc, from which H{K,k) = (K.aok).
D
Proposition 8.2.6 yields bijections
^(/,J)^Cart(^(a/),^(aj)).
We shall generalize this somewhat by replacing S{dJ) with an arbitrary
fibration over S. This yields the so-called "fibred Yoneda lemma"; the
reason for this terminology will become more apparent in the next
section. In the same spirit, the embedding in 8.2.6 will be called the "fibred
Yoneda embedding".
Proposition 8.2.7 Let F: ^ >^ he a fibration. For an object I ^ i,
the fibre J^/ is equivalent to the category Cart(^(9/), J^) of cartesian
functors and cartesian natural transformations from S{dl) to (J^,F).
Proof The proof requires the axiom of choice. Fix an object X G ^i in
the fibre over /. We shall construct a cartesian functor Hx- ^/I ^^
(see 8.2.5):
• for every object (J, a) G ^//, we choose over a a cartesian morphism
a: Xc^ >X in ^ and we put Hx{J^ o^) = X^'-,
• for every morphism 7: (J, a) > (K, /3) in ^/I the relation do'j = a
implies the existence of a unique morphism 7: X^ ^Xp such that
/3 o 7 = a and ^G) = 7; we put Hx G) = 7-
The uniqueness condition in the definition of 7 implies immediately that
Hx is a functor. To prove that Hx is cartesian, using the previous
notation, we choose a cartesian morphism 7': Xa ^Xp over 7 in J^;
the composite /3 o 7' is cartesian over /3 o 7 = ce (see 8.1.4), thus there
exists an isomorphism e in the fibre J^j such that /3 o 'j^ o e = a. By

386 Fibred categories
X. -^^^ X
(Hfh
Ya —^ > Y
ay
Diagram 8.6
uniqueness of 7, this yields 7 = 7^06:, thus the cartesianness of 7. So each
morphism Hx{l) is cartesian and therefore Hx is a cartesian functor.
Fix now a morphism /: X >Y in the fibre J^/. To avoid any
ambiguity, we put indices X, Y in the previous definitions of a, 7. For every
object (J, a) € ^//, the composite / o ax factors uniquely through
the cartesian morphism ay in the fibre over J, yielding a morphism
{Hf)oc\ Hx{J^(y) >HY{J^a); see diagram 8.6. The uniqueness
condition in the definition of {Hf)a implies immediately that those data define
a natural transformation Hf. Hx => Hy which is of course cartesian and
has all its components in the fibres of J^.
The functoriality of the previous construction is clear. Moreover, with
the previous notation, observe that {Hf)ij = /, from which follows the
faithfulness of the process. The same formula will imply the fullness.
Indeed consider a cartesian natural transformation ip: Hx =^ Hy- Its
component on the object (/, 1/) € ^/I is a morphism /: X ^Y which
lies in the fibre J^/, since (p is cartesian. Every object (J, a) € S'/I yields
a morphism a: (J, a) > (/, 1/) in t^/I and thus, by naturality of (p, the
relation foax —oiyo^^. But for such a particular a one has ax = ^x?
so that (^a = {H{f)) by the uniqueness condition defining (i/(/)) .
D
Putting #' = S{dJ) for some object J e S'^ we observe that the fibre
of ^ over I e S' has for objects the morphisms a: / > J, a morphism
7: a >/3 being an endomorphism 7: / >I such that /3 o 7 = a.
Combining 8.2.6 and 8.2.7 thus yields an isomorphism of categories
(f(/, J) ^ Cart(^(a/),^(aj)).
This justifies to some extent our claim that 8.2.7 can be thought of as
some kind of generalization of 8.2.6.

8.3 Fibrations via pseudo-functors 387
8.3 Fibrations via pseudo-functors
Every category S' can be seen as a 2-category where the only 2-cells
are the identities. On the other hand, let us consider the 2-category
Cat of (possibly large) categories, functors and natural transformations
(see 7.1.4, volume 1). It thus makes sense to consider the 2-category
PsFun(^, Cat) of contravariant pseudo-functors, pseudo-natural
transformations and modifications from S' to Cat (see 7.5.1,2,3, volume 1).
Clearly the situation is somewhat simpler than the general setting in
section 7.5 of volume 1, since S' has just trivial 2-cells. The reader who
wants to handle more carefully the questions of size will be better off
using a hierarchy of universes, but such precision is not really relevant
for our purpose.
Theorem 8.3.1 Let ^ be a category. There exists a 2-functor
(f: PsFun(^,Cat) >Fib(^)
with the following properties:
A) given two pseudo-functors P^Q: S' >Cat, if induces an
isomorphism of categories
PsFun(P, Q) >Cart((^(P), (^(g));
B) every fibration F: ^ >i is isomorphic to a fibration (f{P) arising
from a pseudo-functor.
In short, the 2-categories PsFun(^, Cat) and Fib(^) are 2-equivalent.
Proof To understand better the proof which will follow, it is sensible
to start by constructing a pseudo-functor P: S >Cat associated with
a fibration F\ ^ >S. This requires the axiom of choice.
• For an object / € ^, P{I) is the fibre ^j,
• Given a morphism a: J >/ in S we choose, for each object X in
the fibre J^/, a distinguished cartesian morphism otx'- -^a >X.
We must define a functor Pol\ P{I) >P{J)- We put P{a){X) =
Xa. Moreover if /: X >Y is a morphism in the fibre J^/, the
composite / o ax factors uniquely through the cartesian morphism
ay via a morphism X^ >Ya in the fibre over J, which we choose
as P{a){f). Thus P{a){f) is the unique morphism in the fibre over
J such that ay oP(q;)(/) = foax- The uniqueness condition in the
definition of P{a){f) implies immediately that P{a) is a functor.
• Given another morphism /3: K > J in ^, 8.1.4 implies immediately
the existence of an isomorphism P{l3)P{a){X) ^ P{aP){X) in the
fibre over K.

388 Fibred categories
• Choosing in the same way a = 1/ in ^, we get immediately an
isomorphism PA/)(X) = X; indeed the identity on X € ^i is
obviously a cartesian morphism, thus every cartesian morphism over
1/ is isomorphic to an identity.
Considering again lemma 8.1.4, it is straightforward to verify that P is
indeed a pseudo-functor.
With the previous construction in mind, let us now construct a 2-
functor
ip: PsFun(^,Cat) >Fib(^).
Given a pseudo-functor P: i >Cat, we shall thus construct a fibration
G\ ^ >^ whose fibre at / € ^ is precisely the category P[l)'.
• an object of ^ is a pair (/,X) where / € ^ and X € P{1) are
respectively objects of S and P(/);
• an arrow {J,Y) >{I,X) in ^ is a pair (a,/) where a: J >/
and /: Y >P[a){X) are respectively arrows of & and P(J);
• G\ ^ >& is just the first component functor, thus G{I^X) = I
and G(q;, /) = a.
Clearly, we must still provide ^ with a composition which makes it a
category, while G becomes a functor.
Consider arrows {aJ): {J,Y) >(/,X), {C,g): {K,Z) >(J,y) in
^. This yields the following composite in P{K):
Z 9 ,p(^)(y)_P(M/l^p(/3)p(«)(X) = >P{aC){X),
where the last isomorphism comes from the definition of pseudo-functor.
Writing /*^ for this composite, the composition law is defined by the
relation (a,/)o(/3, ^) = (q;o/3,/*^). The associativity of this composition
follows immediately from the first axiom for a pseudo-functor (see 7.5.1,
volume 1). On the other hand the second axiom for a pseudo-functor
indicates that the unit for the composition is equal to the isomorphism
X ~ >P{lj){X) given by the definition of a pseudo-functor. This proves
that ^ is a category while the functoriality of G is obvious. Notice
immediately that the fibre over an object / G ^ is isomorphic to P{I): this
is again due to the isomorphism X = PA/)(X) in the definition of a
pseudo-functor.
To prove that G: ^ >S' is a fibration, let us consider a morphism
a: J >I in S' and an object (/, X) € ^/ in the fibre over /. The pair
{a,lp(a)(x)): (J,P(a)(X)) >{I,X)

8.3 Fibrations via pseudo-functors 389
is thus a morphism of ^. This is a cartesian morphism because given
another morphism in ^,
{X,f):{K,Z) >iI,X),
such that 7 = a o /3 for some C: K > J, the composite /i,
Z ^^PG)(X) =—^P{C)P{a){X),
yields the required unique factorization
(/3,/i): {K,Z) ,{j,P{a){X)).
So G\ ^ >S is indeed a fibration.
Let us verify immediately that starting with a fibration F: ^ ^►^,
constructing a corresponding pseudo-functor P: & >Cat and the
associated fibration G\ ^ >^, the two fibrations are isomorphic. We
freely use the previous notations, without redefining them. We construct
a functor H\ ^ >^.
• If X € ^ is an object in the fibre over /, we define H{X) = (/, X).
• If y € #" is another object in the fibre over J and /: Y >X is a
morphism such that F{f) = a^ f factors uniquely as / = ax o fa
via a morphism fc,: Y >Xct in the fibre over J; we define H{f) =
The functoriality of H follows immediately from the uniqueness
condition in the definition of f^-
Observe first that i/ is a cartesian functor. By construction GoH = F^
since G is the first component functor. On the other hand if / is a
cartesian functor, then both / and ax are cartesian over a with the same
codomain X; the factorization fa is thus an isomorphism. Therefore
H{f) = (a^ fa) is isomorphic to the cartesian morphism (o;, lp(a)(x))
and so is cartesian.
It remains to prove that H is an isomorphism of categories. Obviously,
H is bijective on the objects. To prove it is an equivalence, we must prove
that given X € J^/, Y € J^j, the mapping
^{Y,X) >^{{Y,J),{X,I)), f^iajo,)
is a bijection, where for simplicity we have written a = F{f). This is
obvious since ax is a cartesian morphism.
We must now define
^: PsFun(^,Cat) >F\

390 Fibred categories
on the pseudo-natural transformations and the modifications. Let us
consider two pseudo-functors P^Q: S' >Cat and the corresponding fibra-
tions G: ^ >^, H: Jf >S'. Given a pseudo-natural transformation
9: P =^ Q is giving
• for each object / G ^, a functor 6i: P{I) >Q(/),
• for each arrow a: J >/ of ^, an isomorphic natural
transformation aa- Oj o P{a) =^ Q{a) o 6j^
satisfying the compatibility conditions of 7.5.2, volume 1. Prom (O^a)
we construct now a cartesian functor L: ^ > Jf.
• If {I,X) € ^i is an object of ^, L{I,X) = (/,0/(X)).
• If (a, /): (J, Y) > (/, X) is a morphism of ^, L{a^ f) is the
following composite:
ej{Y) ^'^^^ ,ej{P{a)){X) ^"'^ )Q(a)(g,(X)).
It is routine to check the functoriality of L from the axioms for a
pseudo-natural transformation. One has K o L = G since K and G
are just the first component functors. Moreover the cartesian morphism
(a, lp(a)(x)) of ^ is just by definition applied on (a, cra,x); since aa,x
is an isomorphism in Q{J)^ (a, cra,x) is isomorphic to the cartesian
morphism (a, l(Qa)(x)) via an isomorphism in the fibre J^j and is therefore
an isomorphism.
Conversely if N: ^ >J^ is a cartesian functor, let us prove it arises
from a unique pseudo-natural transformation @, a) via the previous
construction. This construction imposes the relations N{I^ X) = (/, 0/(X)),
N{oiAp{a)iX)) = icx,aa,x) and iV(a,/) = {a,aa,x^Oj{f)) for each
arrow (a,/): (J,y) >{I^X) in ^. This uniquely defines {0,a) in terms
of A^. Since N is cartesian, it maps the cartesian morphism (a, lp(a)(x))
to another; therefore {a^aa,x) is isomorphic to the cartesian morphism
(a, l(Qa)(x)) via an isomorphism in the fibre Jf'j; since the unique
possible factorization is (Ij, cra,x), o^a^x is indeed an isomorphism in Q{J). It
is then straightforward to verify the axioms for a pseudo-natural
transformation from the functoriality of F.
Finally consider another pseudo-natural transformation (r, £:): P => Q
and the corresponding cartesian functor M: ^ >J^. Giving a
modification S: @, cr) => (r, £:) is giving a family Ej: Oj => tj of natural
transformations, for / G ^, with just the requirement that diagram 8.7
commutes since the only 2-ceUs in S' are the identities. One defines ^: L => M
by putting ^(/,x) = ^i,x • Since each S/ is natural, ^ is already natural

8.3 Fibrations via pseudo-functors 391
ejP{a){X) ""''' ^Q{a){eiX)
='J,P{<x)(X)
Q(«)(H7,x)
Diagram 8.7
with respect to all the morphisms in a fibre; the additional axiom for E
to be a modification then yields the naturality of ^, just by definition of
L, M on an arbitrary morphism of ^. The cartesianness of ^ is just the
fact that each morphism ^(/,x) is in the fibre over /.
To conclude, it remains to verify that the previous construction
mapping the modification E to the cartesian natural transformation ^ is
bijective. But giving ^ is giving a morphism ^(/,x) ^^ the fibre J^j over
/ for each object X € P{I)' Putting S/,x = ^(J,x) already yields a
family of natural transformations S/: 9i => tj^ just by naturality of ^. The
additional requirement for S to be a modification is just the naturality
of ^ applied to the cartesian morphism (a, lp(a)(x)) ^^ ^- D
Corollary 8.3.2 Consider an object I of a category &. Via the
correspondence described in 8.3.1, the fibration S{dl) >S of 8.2.5
corresponds with the ''discrete representable functor^^
S{-J): S >Cat
mapping an object J to the discrete category having ^( J, /) as set of
objects.
Proof The fibration is the source functor S/I >S (see 8.2.5), thus
the fibre over J indeed has the arrows J >I as objects and just the
identities J > J as morphisms. Given a morphism a: K ^J and
an object (J, j) in the fibre over J, the corresponding cartesian
morphism is just a: {K^j o a) >(«/,j)- Therefore "the" pseudo-functor
corresponding with S/I >S' indeed acts by composition on the
morphisms. Thus this pseudo-functor is exactly the usual representable
functor ^(—,7): S' ^►Set ... where each set is identified with the
corresponding discrete category. D
Corollary 8.3.2, together with theorem 8.3.1, now explains the
terminology "Yoneda embedding" or "Yoneda lemma" used in 8.2.5 and
8.2.7. For example 8.2.7 indicates via those remarks an equivalence of

392 Fibred categories
categories
PsNat(^(-,/),P) ^P{I)
where P is the pseudo-functor associated with the fibration ^ >^,
^(—,7) is the (pseudo)-functor associated with the fibration S{dl) >S
and PsNat indicates the category of pseudo-natural transformations and
modifications. This is indeed a Yoneda lemma for pseudo-functors.
The fibration S{dl) >S presents the particular fact that it is the
fibration associated with an actual functor S >Cat, not just with a
pseudo-functor. A fibration ^ >S associated with an actual functor
S >Cat thus has the particular property that a "compatible choice of
cartesian morphisms" can be made: for each morphism a: J >I in S
and each object X € J^/, applying P(q;), one can choose a distinguished
cartesian morphism ax'- P{q){X) > (X) in a way which is compatible
with the identities and the composition, i.e. the distinguished cartesian
morphisms ax constitute a subcategory of ^. When this is the case,
^ >S is called a "split fibration".
Definition 8.3.3 A Sbration F: ^ >^ is called a split Gbration
when, for each arrow a: J >I and each object X € ^/, it is possible
to choose a distinguished cartesian morphism ax' X^ > X over a in
such a way that all those morphisms ax constitute a subcategory of ^.
Proposition 8.3.4 Each fibration F: ^ >S is equivalent to a split
nbration G: ^ >S'.
Proof Applying 8.3.1, the fibration F: ^ >& can be described by a
pseudo-functor
P\S ^Cat, I^P{I) = ^i.
Applying our "fibred Yoneda lemma" (see 8.2.7), let us observe that P
is equivalent in the 2-category PsFun(^, Cat) to the actual functor
Q: S ^Cat, / H-> Cart(^(9/), J^)
where the action of Q on a morphism a\ J >/ in S is just composition
with the cartesian morphism &{da)\ 8{dJ) >S'{dI) studied in 8.2.5.
It remains by 8.3.1 to consider the fibration G: ^ >^ associated with
the functor Q. It is equivalent to F\ ^ >& since Q is equivalent to
P. To prove it is split, it suffices to consider the cartesian morphism
(a, l(Qa)(ff)) for each arrow a: J >I in S and each cartesian functor
H G Q(/): these morphisms constitute a subcategory of ^ because Q

8.3 Fibrations via pseudo-functors 393
is an actual functor, thus all the coherence isomorphisms are identities.
D
We have seen in 8.1.9.b that every category ^ induces a fibration
(f: Set(^) >Set. The "dual" of this fibration should be the fibration
(f*: Set(^*) >Set associated with the dual category ^*. Observe that
Set(^) and Set(^*) are by no means dual categories. Indeed consider
a mapping a: I >J in Set and (a, (/^)^e/)*- (Q)z6/ ^{Dj)jeJi a
morphism in Set(^); this is a family (/^i Ci >Da(i))^^j of morphisms
in ^. On the other hand a morphism (a, (^^)^^/): {Ci)i^i >{Dj)j^j
in Set(^*) is a family of morphisms (^gf. Ci ^■^ct(i))i^i i^ ^^^ dual
category ^*, thus a family of morphisms (^gf. Da(i) ^^i)iGi ^^ ^*
Such a family is by no means a morphism {Jj)jeJ ^(C'Oie/ ^^ Set(^)
- except when a is a bijection! This last remark implies in particular
that the fibres of Set(^*) are the dual of the fibres of Set(^). Going back
to 8.3.1 suggests the following definition.
Definition 8.3.5 Consider a Gbration F: ^ >& and a corresponding
pseudo-functor P: S >Cat generating ^ up to an isomorphism (see
8.3.1). The composite
S ^—> Cat —ti—> Cat
where (—)* is the ''dual category functor^^ corresponds by 8.3.1 with a
fibred category G: ^ >^, called the dual fibration of F: ^ >&.
Clearly, the dual fibration is just defined up to isomorphism since the
choice of the pseudo-functor P can only be made up to isomorphism.
Let us conclude this section with a point of terminology, related once
more with the fact that the pseudo-functor P associated with a fibration
is just defined up to isomorphism.
Definition 8.3.6 Let F: ^ >S be a fibration and a: J >I a
morphism in &. We call an ''inverse image functor" a*: J^/ >^j between
the corresponding fibres any functor P{a): P{I) >P{J), where P is
a pseudo-functor associated with the fibration F: ^ >&.
Since two pseudo-functors corresponding with a same fibration are
isomorphic, the inverse image functors a* are just defined "up to
isomorphism" (see 8.3.1). With the previous notations, given an object
X € J^ J we write
ax: a*(X) >X
for a distinguished cartesian morphism. Again, otx is just defined up to
isomorphism in the fibre over /.

394
Fibred categories
F — ^FGF
If * £:
1G * r?
G —^—'-^GFG
£^1g
Diagram 8.8
8.4 Fibred adjunctions
The notion of "adjoint arrows" (see 7.1.2, volume 1) can be immediately
applied to the 2-category Fib(^), for a base category S':
Definition 8.4.1 Consider two fibrations F: ^ >^, G\ ^ >& and
two cartesian functors K: ^ >^, L: ^ >#" in the 2-category
Fib(^). L is a fibred left adjoint to K when there exist two cartesian
natural transformations
rj: 1<^ =^ K o L, e: L o K => 1^
such that the triangles of diagram 8.8 commute.
G:^-
Proposition 8.4.2 Consider two fibrations F: ^
over & and a cartesian functor K: ^ >G. The following conditions
are equivalent:
A) K has a fibred left adjoint functor L;
B) (a) for every object I e S', K: ^i >^/ has a left adjoint functor
Li;
(b) for every morphism a: J >I and every object X e ^i, the
morphism
ax: Lja\X) >a*L/(X),
which we shall describe now, is required to be an isomorphism.
To deGne the morphism ax of condition B)(b), observe that the
morphism
K{aL,x): Ka*Li{X) >KLi{X)
is cartesian since K is a cartesian functor; the composite
a*{X) ^^ >X—n2^^KLi{X),
where r] is the unit of the adjunction Lj H K, thus factors through
KiptLix) via a morphism
ax: a*X >Ka*LiX;

8.4 Fibred adjunctions 395
this yields by the adjunction Lj -\ K the required morphism
ax: Lja%X) >a*L/(X).
Proof The existence of a left adjoint Lj for each fibre means the
existence of a functor Lj: ^/ >#"/ for each fibre, together with
morphisms
r,x: X ^KLi{X), sy: LiK{Y) >Y
in the corresponding fibres, those morphisms satisfying both the
naturality condition with respect to morphisms in the fibres and the triangular
identities for adjunction.
Starting with a fibred adjoint L to K as in 8.4.1, it suffices to restrict
L^rj^e to each fibre to get the required adjunctions Lj H K. Since L is a
cartesian functor, the morphism
L{ax): La%X) >L{X)
is cartesian; by uniqueness (up to an isomorphism) of the cartesian
morphism aL(^x)i the unique factorization
ax: La%X) >a*L(X)
is thus an isomorphism.
Conversely, suppose that conditions B)(a), B)(b) are satisfied and
use the notation of the beginning of the proof. It remains to define the
functor L on all the morphisms and prove the naturality of ry, e with
respect to all the morphisms (not just the morphisms in a fibre). Since
every morphism in a fibration factors as a morphism in a fibre followed
by a cartesian morphism, it remains to handle the case of cartesian
morphisms.
Considering the first part of the proof, we define L{ax) as the
composite
Lja%X) ^^ )a*LKX) ^^^^ ^Li{Xy,
the functoriality of L is immediate. The naturality of rj with respect
to the morphism ax is just the commutativity of diagram 8.9. Notice
that L is a cartesian functor since La'^X is isomorphic to the cartesian
morphism a^jX, thus is cartesian.
To conclude the proof it remains to check the naturality of e or, equiv-
alently (see 3.1.5, volume 1), to prove that the pair {Li{X),r]x) is the
reflection oiX e^ along K: ^ >^. Indeed given /: X >K{J) in
<3, just put J = GKiJ) = F(y), a = G(/): / > J in S. There is a

396
Fibred categories
rjx
ax
^KLiX
a*X-
Va'X
Diagram 8.9
KLa*X
^KLja*X
unique factorisation oi f as f — aK(Y) ° f, for /': X >a*K{Y) in
the fibre ^7. Since K is a, cartesian functor, we get a composite
X-
f
■^a*K{Y)-
-—>Ka*{Y)
which, by the adjunction Lj -\ K, yields a unique factorization in the
fibre ^i, f": Li{X) >a*{Y), and thus the required morphism
Lj{X)-
r
-^a*(y)-
ay
->y.
n
In order to introduce a new useful construction on fibrations, we
"prove" two lemmas.
Lemma 8.4.3 Let S he a category. The category Fib(^) of fibrations
over S has finite products.
Proof The terminal object is the identity functor S==S, The
product of two fibrations F\ ^ >&, G: ^ >S is just their pullback as
functors (see 8.1.15 and 8.1.12). D
The next result indicates that Fib(^) is "cartesian closed up to
equivalences of categories".
—>^, H: Jf >S, There
Lemma 8,4,4 Consider two Sbrations G: %
exists a fibration H^: Jf^ >S' such that for every fibration
F: ^ ^S^ one has equivalences of categories
Proof If the result holds, it must hold in particular when ^ = S{dl)
for some object I ^ S (see 8.2.5). This means by 8.2.7 the existence of
equivalences of categories

8.4 Fibred adjunctions 397
This indicates immediately how to construct Jf"^: it is the split fibration
corresponding with the functor mapping / to Cart(^(9/)x^^, Jf) and
acting by composition (see 8.2.5) on the arrows a: I > J; see 8.3.1.
Now observe that for two fibrations A: si >^, B\ ^ >^,
replacing (j/,A), (^,-B) by equivalent fibrations (see 7.1.2, volume 1) will
transform the category Cart(j/,^) into an equivalent category. Since
the final statement of the present proposition asserts the existence of
an equivalence of categories, we can freely replace each fibration by an
equivalent one. Applying 8.3.4, there is thus no restriction in supposing
that all the fibrations involved in the problem are split fibrations. We
recall that Jf ^, by construction, is already a split fibration.
Going back to the considerations of section 8.3, we shall work with
functors 5, T: & ^.Cart instead of split fibrations. Observe that a
pseudo-natural transformation (see 7.5.2, volume 1) 5 =^ T reduces to giving
• for every object / € ^, a functor Qi\ S(l) >T(l),
• for every morphism a\ J >/ of ^, an isomorphic natural
transformation r^: T{pL) oQi ^ Qj o 5(q;),
in such a way that the following conditions hold:
• if /3: K > J is another morphism of ^,
'Tao/? = T/3 * \s{a) = 1t(/3) * '^a-
Observe finally that performing in Fib(^) the product of two fibrations
A\ si >^, B\ ^ >S is just performing their pullback as functors
(see 8.4.3), thus the fibre {sixs^)i is just the product sij x ^j. So the
product of fibrations in Fib(^) corresponds to the pointwise product of
the corresponding pseudo-functors.
Given two functors Q, R: S >Cat one thus defines a new functor
b9'.S >Cat, /k^ PsNat(^(-,/) xQ,i?)
acting by composition on the arrows of S (see 8.3.2). Given a third
functor P: S >Cat, we must prove the existence of an equivalence of
categories
PsNat(P xQ,R)^ PsNat(P,i?^).
Let us recall that, in terms of a pseudo-functor S: S >Cat,
proposition 8.2.7 is just a pseudo-Yoneda lemma attesting the existence of an

398 Fibred categories
equivalence of categories
5(/)^PsNat(^(-,/),5)
for every object I e S'. Putting 5 = i?^, we have thus already, by
definition of R^, the equivalences
PsNat(^(-,/) X Q,R) ^R^{I) ^ PsNat(^(-,/),i?^),
which is the result in the case P = S'{—^I).
Deducing the result for an arbitrary functor P: & >Cat is rather
straithforward, but a bit lengthy. Since we shall not need the result
explicitly , but just the construction of the fibration Jf^, we give the idea
of the proof and leave the details as an exercise. Given a pseudo-natural
transformation {9^t): P x Q => R^ one defines a pseudo-natural
transformation (^,t): P => R^ by choosing 6j to be the following functor:
P{I) >PsNat((f(-,/)xg,i?), X^ei{X)
ei{X)j: i{JJ) X Q{J) ^R{J), {a,Y)^ej{P{a){X),Y)
for objects I^J^S. Conversely, given a pseudo-natural transformation
(^,r): P =^ i?^, one defines a corresponding pseudo-natural
transformation {6,r): P X Q =^ i? by the formula 9i{X,Y) = ~9i{X)i, where
X € P{I) and Y € Q{I).
An alternative proof of the present proposition would have been to
consider, for every pseudo-functor 5: S >Cat, the equivalence of
categories
PsNat(P,PsNat(y-,5)) ^ PsNat(P,5)
where Y is now the fibred Yoneda embedding mapping / to ^(—, /) (see
8.3.2 and 8.2.6) and the equivalence follows immediately from the fibred
Yoneda lemma (see 8.2.7). In view of 6.6.18, this equivalence presents
P as a "pseudo-Cat-enriched limit" of Cat-representable functors.
Developing the theory of those pseudo-Cat-enriched limits allows us then to
mimic the proof of 6.1.9.d. D
Prom the previous "proof", let us at least extract the important
definition of the exponentiation of two fibrations.
Definition 8.4.5 Let G: ^ >S, H: Jf >^ be two fibrations. The
Gbration H^: Jf^ >S' is by definition the split fibration
corresponding with the functor
S ^Cat, /h->Cart(^//x^^,^)
acting by composition on the morphisms ofS (see 8.2,5),

8.5 Completeness of a fibration 399
With chapter 6 in mind Jf'^ can thus be thought as the fibration of
cartesian functors from ^ to Jff. When ^ is a category internal to (f,
the fibration Jf^^"^) will generally be written Jf"^ (see 8.2.4); this is
intuitively the fibration of Jf'-valued presheaves on ^. When / is just
an object of ^, the fibration jf^(^^) will generally be written J^^ (see
8.2.5); this is intuitively the fibration of /-indexed families in Jff,
Let us conclude this section with a rather obvious "adjoint lifting
theorem" for fibrations.
Proposition 8.4.6 Consider fibrations F: ^ >^, G\ ^ >S and a
cartesian functor K: ^ ^►^ having a Shred left adjoint functor. For
every internal category ^ of S, the cartesian functor K^: ^^ >^^
of composition with K has a Gbred left adjoint functor.
Proof For an object / € ^, Kf is the functor
Can{S/Ix^S{d'S),^) > Can{S/Ix^S{d^), ^)
acting by composition with K. Its left adjoint is just the corresponding
functor LJ acting by composition with L, where L: ^ >^ is a fibred
left adjoint to K. The compatibility condition of 8.4.2 is obvious. Indeed,
given a morphism a: J >/, the functors a* are just composition with
^/ax^id: SjJxsSim) ^SjIx^Sim),
while the functors L^ are just composition with L: ^ >^. Therefore,
given a functor
X'. gjIxsSid^) ^^,
the isomorphism L^ja^'X = a*LjX is just the associativity relation
Lo{Xo {S/ax/\d)) = {LoX)o {S/ax/\d). D
8.5 Completeness of a fibration
Given a fibration F: ^ >8 and a small category ^, we have defined
in 8.1.17 the fibration J^^C^) ^g of diagrams of shape Q) in ^'. the
corresponding pseudo-functor (see 8.3.1) maps an object I e S' to the
category Fun(^, ^/) of diagrams of shape ^ in the fibre ^i; its
action on an arrow a: J >I is just composition with the inverse image
functor a*: ^j >^j (see 8.3.6).
Proposition 3.2.3, volume 1, suggests how to define limits of shape
2 in the fibration ^. It suffices to consider the diagram 8.10, which is
commutative, and the corresponding factorization, still written A,

400
Fibred categories
^ —A-^
F^
A
Diagram 8.10
A: ^■
-^^'
through the pullback in 8.1.17. Observe that A: ^ >^(^) is the
cartesian functor corresponding by 8.3.1 with the pseudo-natural
transformation 5 given by
5/:^/-
-^Fun(^,^/), Xk-^A(X),
and having the identities as coherence isomorphisms. (We assume that
a family of inverse image functors has been chosen once for all.)
Definition 8.5.1 A fibration F: ^ >S has all limits of shape 3), for
a small category S), when the cartesian functor
A: ^
-^^'
has a fibred right adjoint.
As expected, the existence of ^-limits in a fibration is related with
the existence of ^-limits in the fibres.
Proposition 8.5.2 Let F: ^ ^S be a fibration and ^ a small
category. The following conditions are equivalent:
A) the fibration (#", F) has all limits of shape 2);
B) for each fibre ^i^ / € ^, and each functor H: S
-^^I, the limit
of H exists and is preserved by every inverse image functor a*, for
every morphism a: J >I in 8.
Proof This is an immediate consequence of 8.4.2 and 3.2.3, volume 1.
It could appear natural to define the completeness of a fibration
F: 3F ^S by requiring the existence of limits of shape ^, for every
small category Q). But clearly such a definition of completeness would
depend on the consideration of the internal categories V of the category
of sets, while in the theory of fibrations over S one should rather consider
the internal categories ® of S,

8,5 Completeness of a fibration 401
If Si is an internal category of S' and F: ^ >S is a fibration, we
have considered as a special case of 8.4.5 the fibration F^\ ^^ >S
of "^-valued presheaves on ®". We recall that ^^ is a split fibration
whose fibre at / € ^ is given by
(^^)^ ^ Cart(^//x^^(®), J2^).
As in the case of ordinary categories, we want to consider a cartesian
functor
A: ^ >^^
corresponding with the choice of "constant diagrams of shape ®". This
functor must clearly correspond via lemma 8.4.4 with the projection
Let us give an explicit description of it.
For every object / € ^, we define a functor
A/: ^i >Cd^rt{S/IxsS{S),^)
in the following way. An object X € ^i is mapped to the composite
S/IxsS{3)) >S/I >^,
where the first functor is just the projection and the second functor
corresponds with X G ^i by the pseudo-Yoneda-lemma (see 8.2.7), thus
it maps a\ J >I to q;*(X). This construction extends immediately
to the arrows /: Y >X of .^//, just by considering the morphisms
a*{f). When #" is a split fibration, the functors A/ are natural in /,
thus define by 8.3.1 the required cartesian functor A. When ^ is an
arbitrary fibration, the functors A(/) are just pseudo-natural in /, but
this again yields the required functor A.
Definition 8.5.3 Let F: ^ >S be a Gbration and Si an internal
category oiS, The Gbration {^^F) has all limits of shape S) when the
cartesian functor
A: J^ >J^^
has a Gbred left adjoint.
Given a fibration F: 3F ^8^ one might now want to define its
completeness as the existence of all limits of shape ®, for all internal
categories 3) of S. This is not enough, for two reasons. First of all, without
any further assimiption on S, there would not be any reason to find in S
some particuTar internal categories ® which would exhibit the existence

402 Fibred categories
of "basic" limit diagrams like puUbacks, equalizers, and so on. The
second reason is more subtle and is related to the need for good stability
conditions of the notion of completeness. We explain this by an example.
Suppose ^ is a complete category, in the usual sense. Obviously a
good notion of completeness must imply that the canonical fibration
Set(^) >Set of 8.1.9.b is complete, the limit of families being
computed pointwise. But fix now a set /. One could consider an even more
sophisticated category Set/(^) where the indices are now the families
{Ji)iei of sets and an object of ^ is thus a family of families of objects
of ^: ({Cji. j_) .^r. It is easily observed (we shall do it immediately for
an arbitrary fibration) that we get in this way a new fibration
Set/(«^) ^Set//
where Set// is the category of /-families of sets (see 1.2.7.a, volume 1).
Clearly this new fibration should again be complete, with the limits of
families of families still computed pointwise. Such a stability condition
of the notion of completeness turns out to be important in practice and
must be included in the definition.
Definition 8.5.4 Let F: ^ >S be a Gbration and I ^ S an object.
The locahzation of F at I is the Gbration
F{i)'^{i) >^II
whose Gbre at a: J >I is just ^i and whose inverse image functor
along 7: (J, a) > (/C, /3) is just 7*.
It is obvious that F(/) is a fibration. We are now ready to define the
completeness of a fibration.
Definition 8.5.5 Let F: ^ >S be a Gbration, where & is a category
with Gnite hmits.
A) This Gbration is Gnitely complete when it has all limits of shape 3),
for every Gnite category S);
B) this Gbration is internally complete when it has all limits of shape
Si, for every internal category Si in S';
C) this Gbration is complete when, for every object I e S, the Gbration
F{i)'' ^(i) >S/I is Gnitely and internally complete.
Proposition 8.5.6 A complete Gbration over a base category with Gnite
limits is Gnitely and internally complete.
Proof Put / = 1 (the terminal object) in 8.5.5. D

8.5 Completeness of a fibration 403
J X / ^K X /
PJ
Pk
J —oT-^ K
Diagram 8.11
We shall now give a striking characterization of complete fibrations
(see 8.5.9), which is also Benabou's original definition of complete
fibrations. We split the proof into several lemmas.
Lemma 8.5.7 Let F: ^ >& be a Gbration, with & a Gnitely complete
category. The following conditions are equivalent:
A) the Gbration (#", F) has all hmits of shape dl, for every object I e S'
(see 8.2.5);
B) for every object J ^& the inverse image functor p'ji ^ j >^ ixJ
has a right adjoint Ylj such that, for every morphism a: J >K,
the comparison morphism a* o J|^ =^ Ylj o (a x 1/)* is an
isomorphism (see diagram 8.11).
Proof Let us first make explicit the form of the "comparison
morphism" in condition B). Given an object X € ^ixK-, the counit of
the adjunction p|^ H Hk gives a iiiorphism ex'- PkIIkC"^) ^"^- "^^^^
yields the composite
P^^IIkW >{c^xhrP*K]lr,(^)^'' "" ^'^*''' (IpxaYiX)
which corresponds via the adjunction p} H Ylj with the required
morphism o;*!!^!"^) ^rijC^ ^ l/)*(-^)- Moreover observe that we can
replace the fibration {^^ F) in the proof by a split fibration; in view
of 8.3.4, there is thus no restriction in supposing that {3^^F) is a split
fibration. In this case p'jOt^ = (a x l/)*p|^.
Let us now make explicit condition A). It means the existence of a
fibred right adjoint to the functor A: ^ >^^. The fibre at J € ^ of
the fibration ^^ is (see 8.4.5, 8.2.5 and 8.2.7)
C^^) J ^ Cart(^/Jx^^//, J^) ^ Cart(^/J x 7,^)
For a morphism a: K > J, the inverse image functor a* of ^^ is
just the inverse image functor (a x 1/)* of ^. The cartesian functor

404
Fibred categories
L > K
0
a
-> I
Diagram 8.12
{L,ao6) = {J,a)x{K,0)-
7
-^(/,l/)x(K,/3) = (K,/3)
0
Diagram 8.13
A is defined by the fact that Aj: ^j >^jxi is the inverse
image functor along the projection pj: J x I >J. The existence of a
fibred right adjoint to A thus means the existence of a right adjoint
rij- ^Jxi >^J for each object J € ^, together with a natural
isomorphism a* o Ylj^ = Ylj o (a X 1/)* for every morphism a: K >J in
S (see 8.4.2), thus exactly condition B). D
Lemma 8.5.8 Let F: ^ >8 be a Gbration, with 8 a Gnitely complete
category. The following conditions are equivalent:
A) for every morphism a: J >I in S, the Gbration
has all limits of shape d{a);
B) for every morphism a: J >I in 8 the functor a*: #"/ >^j
has a right adjoint Yl^ such that, for every puUback in 8 as in
diagram 8,12, the comparison morphism a* o H/? =^ 117 ° ^* ^^ ^^
isomorphism.
Proof This is an immediate consequence of 8.5.7, just observing that
the pullback in condition B) can equivalently be written as in diar
gram 8.13, in the category <^//, where the product is just the pullback
over /. D
Theorem 8.5.9 Let 8 be a category with Snite limits, A Gbration
F: y >S is complete if and only if:

8.5 Completeness of a fibration 405
L > K
P
> /
Diagram 8.14
A) the fibration {^^ F) has all limits of shape ^, for every finite
category Q)]
B) for every morphism a: J > I in S, the corresponding inverse image
functor a*: #"/ >^j has a right adjoint Yl^;
C) for every puUback in S^ as in diagram 8.14, the comparison morphism
^* ° 11/3 ^ 117 ° ^* ^^ ^^ isomorphism.
Proof For each object I e S', the fibration F(/): ,^(/) >S'/I is built
up from the fibres of ^ and the inverse image functors of ^ (see 8.5.4);
thus if in ^ each fibre is finitely complete and each inverse image functor
preserves finite limits, the same holds in «^(/). Applying 8.5.2, this isjust
saying that the finite completeness of the fibration (,^, F) implies that
of each fibration (,^(/),F(/)). The converse is obvious, putting / = 1 in
8.5.5 (see 8.5.6).
Lemma 8.5.8 indicates that a complete fibration (,^, F) satisfies
conditions B), C) of the present theorem. So it remains to prove that
conditions A),( 2), C) in our statement imply the internal completion of
each fibration (^(/),F(/)). This is the fibred version of theorem 2.8.1,
volume 1. But observe immediately that the validity of conditions A),
B), C) of the statement for a fibration F: 3^ >S immediately
implies the validity of the same conditions for each localized fibration
F{i)\ ^{i) >^Ih with I e S'; this is due again to the fact that ,^(/)
is built up from the fibres and the inverse image functors of ^. Thus it
suffices to prove that conditions A), B), C) of the statement imply that
^ has all limits of shape ®, for every internal category Si of S'. This
fact, applied to each fibration F(/): .^(/) ><^/I^ is precisely condition
B) in 8.5.5.
With the notation of 8.1.1, volume 1, let us fix an internal category
® = {Do,Di,do,di,i,c) in S'. We must produce a fibred left adjoint to
the cartesian functor A: ^ >^^. Let us compute it explicitly. For

406 Fibred categories
every object / € ^,
the inverse image functors acting by composition (see 8.2.5). We must
still compute the form of the fibration S/IxgS{D). Given an object
J ^ S^ its fibre at J can be described as follows (see 8.1.10):
• the objects are the pairs (j, m), where j: J >I and m: J >Dq\
• a morphism /: (j, m) > (fc, n) in the fibre over J exists just when
j = fc, since the fibres of S/I are discrete categories (see 8.3.2); a
' morphism /: (j, m) > (j, n) is then a morphism /: J >Ki in ^
such that do o / = m, di o / = n.
This fibre of S/IxgS{Qf) at some object J € ^ can thus be described in
an equivalent way:
• the objects are the morphisms a: J >I x Dq (put a = (j^m) in
the previous description);
• a morphism 7: a >/3 is a morphism 7: J >Di such that
A/Xdo)o7 = a, A/xdi)o7 = /3;
put 7 = (j, /) in the previous description; composing the identities
with the first projection yields j = k and with the second projection,
do o f = m, di o f = n.
Given the internal category Si and the object /, multiplying by / all the
data constituting ®,
(/ X Do,I X Di,l/ X do,l/ X di,l/ x i, 1/ x c)
yields a new internal category which we will denote I xQ). Observe that
considering the first projections / x Do >I, I x Di >I, ... those
data can also be seen as an internal category in S/I, which we shall
denote ®//. We have then the equivalences
J^f ^ Qm{SlIxsS{&),^) ^ Cart(^(/ x &),^)
- Cart(^//(®//), J^(,)) - (J^(/))f/',
proving that the fibre of ^^ at / is just the fibre of {^{i))^' at the
terminal object 1/: / = / of S/I. On the other hand the fibre of #" at /
is precisely the fibre of #'(/) at the terminal object 1/ of S/I.
Constructing a fibred right adjoint V: ^^ >^ to the cartesian
functor A is constructing a right adjoint V/: ^f >^i to A/, for

8.5 Completeness of a fibration 407
each object / G ^, together with compatibility conditions (see 8.4.2).
Our previous description of #'f indicates that we must construct a right
adjoint
Vi,: (i^//)f/' 'i^/I)u
to Aij, where 1/ is the terminal object of S'/I. Again since the conditions
A), B), C) of the statement are stable under localization at an object
/ G ^, it suffices to prove the existence of Vi, for 1 G ^ the terminal
object, and apply this result to #'// and &/I to get the existence of
V/.
Thus we must now consider the functor
Ai: ^i >Cart{S{S),^)
mapping an object X G ^i to the "constant cartesian functor on X",
i.e. the functor
Ai{xy.s{&) >^
mapping Y G S'{Si)j to a*j{X)^ with aj: J >1 in S. We must
construct a right adjoint Vi to Ai. It suffices to mimic the proof of 8.1.1,
volume 1. Let us consider a cartesian functor H: S{Si) )►#'; we must
define an object Viif G ^i.
The identity 1£)q: Dq Dq is an object in the fibre S{Si)do' Its
inverse images in S'{S)di^ along the morphisms do^di: Di ^Dq^ are
just do: Di >Do^ di: Di >Do (see 8.1.10). On the other hand in
the fibre ^{31H1^ the identity on Di yields a corresponding morphism
(Di,do) ^►(Di,di), since do o 1^^ = do and di o 1^^ = di. Applying
the cartesian functor if, we thus find an object Hq G ^Dq-, the image of
Ido ^ ^(®)di. Since H is cartesian, d^{Ho) = H{Di,do) and dj(ifo) =
H{Di^di). Therefore H{1di) yields a morphism (p: dQ{Ho) >di{Ho)
in the fibre ^Di •
To understand the above intuitively consider the case where ^ is the
fibration Set(^) for some complete category ^, as in example 8.1.9.b.
Then ® is a small category. Given a functor H: Q) ^►^,
while (^ = (-ff(^))^g£, • As will be observed in 8.5.9, given the mapping
Oil'. I ^1 in Set, the corresponding adjoint functor
JJ/ Set{^)j >Set(^)i
of a} maps the family {Ci)i^i to Yli^jCi.

408 Fibred categories
Let us come back to our general construction. Given the morphism
a/: / >1 in ^, we write H/ ^^^ *he right adjoint to a}. As in 2.8.1,
volume 1, consider first the two objects YIdq{Ho) and YlD^dKHo) in the
fibre ^i. We shall construct a, 6: Hpo(-^o) >Hp^^i(-^o) ^^d define
Ah- Vi(if) >YId (^o) ^ ^^^ equalizer of a, 6 in ^i. The morphism
a corresponds via the adjunction a})^ H Hdi ^^^^ ^^^ composite a:
where £: is the counit of the adjunction a})^ H Hdo * ^^ ^^^ other hand
6 corresponds via the adjunction a})^ H Hdi ^^^'^ ^'^^ composite 6:
To prove that Vi {H) is the coreflection of H along Ai, we must
construct a cartesian natural transformation th'- AiVi(if) =^ H. Let us
consider an object (/, m) G ^(®); we must define in the fibre ^i a
morphism Ai Viif (/, m) > Jy(/, ^), thus by definition of Ai a morphism
a|(Viif) ^ if (/, m). But in the fibration ^(^), (/,m) = m* (Do, Ido);
therefore H{I^ m) = q.'^Hq^ since H is cartesian. The required morphism
is then the composite
and it is routine to verify the naturality. Next, given an object X G ^i
and a cartesian natural transformation 9: Ai(X) => if, we must find a
unique arrow x: X >Vi{H) in ^i such that r^o Ai(x) = 0. Finding
X is equivalent to finding y: X > Hdq i^o) in ^i such that aoy = boy.
The morphism y corresponds, via the adjunction a})^ H Ylj-^ , with the
composite
Applying the adjunction aJ,^ H YIdi> ^^^ equality aoy = boy follows
from consideration of diagram 8.15. Each internal piece of the diagram
is by definition commutative; thus the equality aoa^^^y = boa'j^^y will
follow from the commutativity of the outer diagram. This is the case by
naturality of 0, since
do0(Do,lDo) = ^d*{DoAo) = ^{Dudo)^
dl0(Do,lDo) = ^dI(Do,lo) = ^{Dudi),

8.5 Completeness of a fibration
1
^
1
doO{DoADo)
Diagram 8.15
tdlHo
f
<^Do^ ^^o'^Doi.i.Do^O ^ do-no
J
409
This concludes the proof that the functor Ai: J^i ^Cart(^(®), ^)
has a right adjoint Vi. We have already observed that, considering the
fibration F(/): .^(/) >S'/I and the internal category Si/I of ^//, this
implies that every functor A/: ^/ > Cart(^//x^^(®), J^) has a right
adjoint V/. It remains to prove that the family (V/)/ of those right ad-
joints satisfies the compatibility condition of 8.4.2. This is not at all
a "straightforward consequence of the naturality of the constructions":
this fact depends on condition C) of our statement. Let us first make
explicit the construction of V/. Let H: S/Ix^S{Sf) >^ be a
cartesian functor. Just reorganizing the fibres over ^//, this yields a cartesian
functor H: S/I{3}/I) >^/I. Let us now write Pq: I x Dq ^/ and
p{: I X Di >I for the first projections of the products defining IxQ).
The object V/(if) G ^i is obtained via an equalizer
VK^)-
Ai
H
ai
bi
Since the inverse image functors of ^ preserve equalizers (see 8.5.2),
the compatibility condition required for the functor V/ will be implied
by the analogous property for Opo^ Opi^ due to the naturality of the
construction of a/, 6/. Given a morphism /3: J >I in <^, it sufiices to

410 Fibred categories
J X Do >I X Do J X Di >J X Di
Pi
Pi
-^ I
Pi
13 ■ ^ ^
Diagram 8.16
P{
■^ I
consider the puUbacks of diagram 8.16 to get, by condition C) of the
statement, the required isomorphisms /?* o J^^/ ^ fj^j o (/? x Ido)*,
/?'onp{=npf°(/3xiDj*. ° " □
To conclude this first approach to limits in a fibration, it remains to
verify that for ordinary categories, fibred completeness reduces to the
usual completeness.
Proposition 8.5.10 Let ^ be a category. The following conditions are
equivalent:
A) ^ is complete;
B) the fibration Set(^) ^►Set is complete.
Proof The completeness of the fibration Set(^) ^►Set implies the
usual completeness of each fibre (see 8.5.2), thus in particular the
completeness of the fibre over 1, which is ^.
Conversely if ^ is complete, for every set / the category ^^ of I-
indexed families of objects and arrows in ^ is complete and limits are
computed pointwise. Given a mapping a: J >I between two sets,
the corresponding inverse image functor a*: ^^ >^'^ maps a
family {Ci)i^i to the family (C^y)) .^j, thus preserves limits since these are
computed pointwise.
Consider now a mapping a: J ^I in Set and two families {Dj)j^j^
{Ci)i^i of objects of ^. Let us define
a
>*', ww"(n„.-.,.,«>),^,
and analogously on the morphisms. A morphism
in ^if*^ is an /-indexed family of morphisms fi: d ^Iljea-iCi)^^' ^^
equivalently a J-indexed family of morphisms gy. Ca{j) >Dj: just put
ei

8.5 Completeness of a fibration
L ~—> K
0
411
a
-> J
Diagram 8.17
Qj = pj o fi if a{j) = i. This proves that Ha '^^ right adjoint to a*.
Finally we must consider a puUback in Set, as in diagram 8.17, and
a family {Xj)j^j of objects of ^. We must prove the isomorphism /3* o
Ua = U-,oS*. Indeed
keK
k€K
To conclude the proof it remains to observe that
{j\jeJ, jea-^0{k)} = {j\jeJ, a{j) = m}
= {6il)\leL, jil) = k}
= {6{l)\leL, lej-Hk)}
just because the puUback L is defined by
L={{j,k)\jeJ, keK, a{j) = m}'
n
The construction of the functors Yl^ in 8.5.10 and the fact that their
existence is equivalent to the existence of limits defined on discrete
internal categories (see 8.5.8) justifies the following definition
Definition 8.5.11 Let F: ^ >S' be a Gbration, with S a Gnitely
complete category. This Gbration has ^-products when:
A) for every morphism a: J > I in 8 the functor a*: #"/ > ^ j has
a right adjoint Yl^;
B) for every puUback as in diagram 8.18, in S, the comparison morphism
^* ° 0/3 =^ 07 ° ^* ^s an isomorphism.
Finally one defines

412 Fibred categories
L > K
/?
Diagram 8.18
Definition 8.5.12 Let ^ be a category with Gnite limits.
A) A Sbration F: ^ >S is cocomplete when the corresponding dual
nbration F*: J^* >S' (see 8.3,5) is complete.
B) A Gbration F: ^ >S has S-coproducts when the dual Gbration
F*: J^* >S' has (^-products.
8.6 Locally small fibrations
An essential feature in the definition of a category ^ is the requirement
that the morphisms between two objects constitute a set. This is indeed
crucial for proving, for example, the Yoneda lemma, on which many key
results depend. When studying the fibrations over a base category ^, the
"indexing objects" are now those of S'. In some sense, S' now replaces the
category of sets for many purposes. Therefore a special attention should
be paid to those fibrations F: 3^ >8 for which "the arrows between
two objects can be represented by an object of ^".
To understand the definition better, let us consider again the basic
example of the fibration Set(^) ^►Set, for a small category ^ (see
S.l.Q.b). Given two objects C, D in ^, i.e. in the fibre of Set(<i?) over 1,
we are interested in considering all the arrows from C to D. This yields
the set ^(C,D). Now considering the mapping a: ^(C, D) )►!, we
can consider the two objects ol^C, ol'D in the fibre over ^{C^D)\ they
are just the constant families (C)/ie<^(c,D)> {P)he^{C,D)' Between those
two families, we have a very canonical morphism /: ot'iC) ^►a*(D),
namely
(/i: C ^D)h^<^(^c,D)
i.e. the family of all arrows from C to D. That this construction is
canonical is attested by the following universal property. Consider a set J, the
corresponding mapping /?: J >1 and a morphism g: P*C >I3*D.
This arrow ^ is a family of morphisms {qj: C >D)j^j. This deter-

8.6 Locally small fibrations 413
mines the unique mapping
7: J >^(C,D), j^gj
such that 7*(/) = g. We have just expressed the representability of the
morphisms between two objects C, D of the fibre over 1. It remains, by
localization, to express the same property for two objects in an arbitrary
fibre.
Definition 8.6.1 Let F: ^ >8 be a Gbration. This Gbration is
locally small when, for all objects I E S and X^Y E #"/, it is possible to
Gnd
A) an object ^{X^Y) in 8 and a morphism axv- ^{X^Y) >I,
B) a morphism fxY- o^xvi-^) ^^xyO^) ^^ ^^^ ^^^^ ^^^^ «^(-^>5^)?
these data being universal for these properties, which means that given
A) an object J in S and a morphism C: J >I,
B) a morphism g: /3*(X) >/3*(F) in the fibre over J,
there exists a unique morphism 7: J >^{X, Y) such that axy 07 = /3
andYifxv) = g-
It is worth emphasizing here the fact that our axioms for a category
already contain the requirement of "set-theoretical local smallness", i.e.
the fact that the morphisms between two objects constitute a set. So in
definition 8.6.1, the requirement is just that of "internal (f-smallness"
(compare with definition 8.5.5).
Observe that giving axv- ^{X, Y) >I is precisely giving an object
in S'/I, as suggested by the considerations preceding the proof.
Our first observation must obviously be
Proposition 8.6.2 Let ^ be a category. The fibration Set(^) ^►Set
is locally small.
Proof Consider a set / and two families of objects X = {Xi)i^i,
Y = {Yi)i^i. The set ^{X, Y) is not the set of morphisms from X to F
in Set(<i?) (how would it be possible to "fibre" it over /?); it is
j^(x,y) = II.^^^(x,,y,)
with axY'- ^{X, Y) >I mapping a morphism h G ^{Xi, Yi) to i e L
The morphism fxy: a*xY{X) >a*xY{Y) is then the family of
morphisms
{h: Xi ^^*)/i€]J.^^<^(Xi,yo-

414 Fibred categories
Next, given /3: J >I and g: l3*{X) >/3*(y), the required mapping
7: J >^{X^ Y) maps the index j G J to gji X^(^j) ^^/30)- ^
Now, a result which one should expect if the terminology is
appropriate:
Proposition 8.6.3 Let (f be a finitely complete category. Every small
Gbration over S is locally small. In other words (see 8.1.11), for every
internal category ^ of S, the Ghration S{9S) >S is locally small.
Proof Consider first two objects A, m), A, n) in the fibre of S{^) over
1. We must construct in S the object ^{m^n) of "arrows of domain m
and codomain n". Clearly the object ^(m, •) of all arrows of domain m
should be the equalizer
dp
^(m,#)> >Ci Co,
mo7
where 7: Ci ^►l; in the same way the object ^(•,n) of all arrows of
codomain n should be the equalizer
'i?(.,n)> .CiZZ=tCo,
and finally ^(m, n) should be the intersection
^(m, n) = ^(m, •) D ^(•, n)
as subobjects of Ci.
We must now adapt this intuitive construction to the case of two
objects (/, m), (/, n) in the fibre of S'{^) over I e S'.In other words, we
must perform the previous construction over the terminal object in the
localized fibration F(/): #"(/) >S'/I. We thus consider the equalizers
1/ X dp ^
^(m,#)> >I X Ci ~tT X Co,
(l/,mopj)
1/ X di
^(•,n)> >I X Ci ^/ X Co,
(l/,nop/)
where pi: I x Ci >/ is the first projection. Then ^{m^n) is defined
as the intersection
^(m, n) = ^(m, •) D ^(•, n)
as subobjects of J x Ci. The morphism amn- ^(m, n) >I is the
composite
^{m,n)) ^ >I X Ci 2Z—^/.

8.6 Locally small fibrations
IxCo ^""^ ) Co
ih^npi)
1/ X do do
^{m,n)> ^ >/xCi-
PCi
P/
^ Ci
415
Diagram 8.19
We must still define the generic arrow fmn'- (^mni^^ ^) ^^mni^^ ^)*
This is the composite
^{m,n)> ^ >I X Ci ^^^ )Ci.
Indeed,
<^mnU>^) = (^(m,n),moa,nn), a;;;,^(/,n) == (^(m,n),no a,nn).
Moreover, considering diagram 8.19, we get the relation
do o fmn = doopc^oi= pc^ o (Ij X do) o i
= PCo o A/)i^Pi) oi = mopi ox
= moOLmn'
In the same way one gets di o /^^ = noa^n> proving that the morphism
fmn'- a^^ni^.m) >a'^^{I,n) is indeed in the fibre over ^{m,n).
It remains to check the universality of our construction. Let us consider
morphisms /3: J > I ills' and g: /3* (/, m) > C* (/, n) in the fibre over
J. The latter is thus a morphism g: J >Ci in S such that do^^ = rno/B^
di o g = no /3. Considering diagram 8.20, we find at once
A/ X do) o (/3,^) = {C,doog) = (/3,mo /?) = (l/,mopj) o (/3,^),
from which we get a unique factorization of (/3, g) through the equalizer
^(m, •). Analogously, the relation di og = no/3 implies that (/3, g) factors
through ^(#,72), from which finally {l3,g) factors through ^{m,n) via a
unique morphism 7: J >^{m,n). One immediately has
Q^mn o -y = pj o i o h = pi o {i3, g) = l3.

416 Fibred categories
J
@.9)
1/ X do
^(m,#)> ^/xCiZZZIIZZlt ^xC'o
(l/,mopj)
Diagram 8.20
On the other hand,
proving 7* (/mn) = 9-
The uniqueness of such an arrow 7 is attested by the relations
Pj o i o 7 = am,n oy = C,
which imply 207= (/3,^), with i a monomorphism. D
In 1.3.2, volume 1, we observed that the functors and natural
transformations from a category ^ to a category ® again constitute a category -
when ^ is small! This smallness condition was necessary to ensure that
there is just a set of natural transformations between two functors from
^ to S>. This result carries over to the case of fibrations. We prove first
two useful general lemmas:
Lemma 8.6.4 IfF: ^ >S is a locally small fibration, then for every
object I e S', the localized fibration F(/): #"(/) >S'/I is also locally
small.
Proof For an object j: J >I of S/I and two objects X^Y in the
fibre of #"(/) over {J,j), X, Y are in fact two objects in the fibre #'j of
#". Thus we get by assumption the morphisms
axY:^{X,Y) >J, fxy: a^^yiX) >a3,y(y);
see 8.6.1. This immediately yields the morphism
axy: (J^(X,y), j o a^y) >{J^J)
in <?/J, from which the result follows. D

8.6 Locally small fibrations 417
^{X,Y)xi^{Y,Z) ^^^ )J^(y,Z)
Pxy\ \olyz
Diagram 8.21
Lemma 8.6.5 Let F: ^ >S' be a locally small Gbration on a category
S with finite limits. For all objects I E S and X^Y^Z e #"/, there exist
a ^^composition morphism^^
IXYz: ^{X,Y)xi^{Y,Z) >^{X,Z)
and a ^^unit morphism^^
ex: I >J^{X,X)
presenting the category ^j as a category enriched over the monoidal
category S/I provided with its cartesian product as a multiplication
(see 6.2.1).
Proof Let us thus consider in S' the puUback of diagram 8.21, i.e. the
product in S'/I, and write axvz = olxy o pxv = otyz o Vyz- By the
local smallness of the fibration, the consideration of olxyz together with
the composite
yields the required composition morphism
IXYz: ^{X,Y)xj^{Y,Z) >^{X,Z)
such that axz o 7xyz = c^xyz and
^XYzifxz) = PyzUyz) O P*XY{fxY)'
For the unit morphism ex^ it suffices to consider the arrows
1/: / >/, Ix: X >X.
The universal property for local smallness implies the existence of a
unique morphism ex'- I >^{X,X) such that
otxx o£x = 1/, e^'xUxx) = Ix-
This already describes the properties of the composition and unit mor-
phisms in terms of the fibration. Prom this it is routine to conclude the

418 Fibred categories
proof, an exercise which is left to the reader since the corresponding
properties will not be needed in this book. D
Theorem 8.6.6 Let S he a, finitely complete category such that the
cajionical Gbration oiS over itself,
Si: Ar((f) >(f
(see 8.1.9.C), is complete. If F: ^ >S and G: ^ >8 axe two fibra-
tions over 8 with (^, G) small and (.^, F) locally small, the fibration
pG. ^"S ^g jg again locally small.
Proof Both the notions of completeness and of local smallness are
stable under localization (see 8.5.5, 8.6.4). Therefore it suffices to prove
the local smallness condition for two objects in the fibre over 1; the
general statement for two objects in the fibre over I E S' then follows by
working in the fibre over the terminal object 1/ in S/I.
Since a fibration equivalent in Fib(^) to a locally small one is obviously
locally small, there is no restriction in supposing that F: ^ >S is a
split fibration (see 8.3.4). We consider then an internal category ^ of
S and take (^,G) to be the fibration g(^) >S of 8.1.10. We must
prove that the fibration ^"^ ^8 (see 8.4.5) is locally small. As already
observed, it suffices to prove the required condition for two objects in
the fibre over 1. Recall that the fibre of ^^ over an object / G (f is
^"i ^ Cart((f(/ X "S),^) ^ Cart((f//(^//),^(j));
see the proof of 8.5.9.
We consider two objects K, L in the fibre ^f, i.e. two cartesian
functors K, L: S{^) \^, Again as observed in the proof of 8.5.9, K
determines an object Kq G ^Cq and a morphism k\ do(^o) "^^VJ^^
in the fibre ^Cx • I^ ^he same way L determines an object Lq G ^Cq
and a morphism A: d5(I/o) ^c?i(Lo) in the fibre ^c^. The reader will
again refer to the proof of 8.5.9 to remind himself of the intuition lying
behind these data.
The assumption of completeness of the canonical fibration of 8 over
itself implies, for every arrow a: J ^I in (f, the existence of an adjoint
pair of functors a"*" H J][^,
a+: Sjl >SIJ, TT : SjJ-
where a"*" is just pulling back along a; see 8.5.9. (We have written a"*"
instead of a* to avoid confusion with the inverse image functors in

8.6 Locally small fibrations 419
^.) When / = 1 is the terminal object, there is a unique morphism
a: J >1 and we generally write Ylj instead of Ha-
Intuitively, a morphism C: K > J can be seen as a family {Kj)j^j\
see 1.2.7.C, volume 1. The object YIA^^P^) ^ ^ 1^ - ^ should then be
thought of as the product rijeJ^J* Iiituitively again, a natural
transformation 0: K =^ L is a Co-indexed family of morphisms
{Bv:K{U) >i(f/))f;,Co'
i.e. an element of Hco^^^^O' ^^' "^^^ object of natural transformations
K => L should thus be a subobject of Ylco^i^^^ -^o)*
Let us consider first the objects dQ{Ko), di{Ko) in the fibre ^Ci- By
local smallness of #", we get universal morphisms
qk: ^{dl{Ko),dl{Ko)) yCi, Ik: a*j,d*o{Ko) ^aJ,dI(Ko).
Considering then the morphisms
lc,:Ci >Ci, K-.dUKo) >dl{Ko),
we find a unique factorization P^: Ci ^►#'(^0(^^0M di{I^o)) in ^ such
that aK^^K == Ici and P^kUk) = i^- We can view all this as a morphism
/3if:(Ci,lcJ >(^{dl{Ko),dl{Ko)),aK)
in S/Ci. Considering the unique morphism Si: Ci >1, the
corresponding functor
maps 1 to e'i{l) = (Ci, Ici)- Writing Yl^^ for the right adjoint of this
functor, Pk corresponds by adunction with a morphism
-^^: 1 ,l[^^(^^{d*o{Ko),dl{Ko)),aK).
In the same way one constructs
■0^: 1 >ll^^(^^{dl{Lo),dl{Lo)),aL).
Intuitively, Yl^^ (^^{dQ{Ko), di{Ko)),aK) is the set of all families of
morphisms indexed by Ci; l3j^ picks up the generic family {u)ueCi'
Our next observation is quite general. Consider two objects X, Y in
the fibre ^j and the corresponding universal data deduced from the
local smallness of (^, F):
axYi ^{X,Y) >/, /xy: a3^y(X) ^^xrC^-

420 Fibred categories
6*axY
axY
J —^-^ I
Diagram 8.22
Given an arrow 6: J >I in (f, we can consider the objects 5*X, 5*y
in the fibre ^j, together with the universal data
Considering in S' the puUback of diagram 8.22, i.e. applying the inverse
image functor 6'^ in the canonical fibration of S' over itself, we can
consider the morphisms
S+iaxY): 6+{^{X,Y)) >J,
rifxr): Ta*xy{X) >ra*xy{Y).
Since 6 o a'^y = {6'^{axY)) o 5*, this yields a unique factorization
6: E+(J^(X,y)) >^{6%X),6%Y))
such that 5*(axy) = Q;^*(x)/*(yH<5 and^ (/xy) = <5*(/<5). In particular
we have a morphism
6:6+{S^{X,Y),axY) >(.rF*(X),6*(F)),a,.(x),6.(y))
in the fibre over J, which corresponds by the adjunction 6'^ H Yl^ with
a morphism
6: {^{X,Y),olxy) —
in S/I, We can further apply Hj to get
Yii^-^j{nX,Y\axY) >\{j[s^{8*{X),6*{Y)),ae.(x),6'(Y)).
We shall in particular apply this construction with 6 = do ox 6 =
di, choosing X = Kq and Y = Lq, Intuitively, fJc ^ niaps a
family (tt;: if(C/) >L{U))^^^^ to the family (%: ii:(C/) >L{U))^^^^,

8.6 Locally small fibrations 421
where u: U >V. In the same way Hco^i maps the same family of
morphisms to (ty. K{V) ^H^))uec '
We are now ready to define the object ^^{K^L). For the sake of
brevity, we shall now write Yli^{X, Y) instead of Yli{^{X, F), axr),
when no confusion can occur. By lemma 8.6.5 we get a composition
morphism
■yK:^{d*o{Ko),dl{Ko))xc,^{dl{Ko),dl{Lo)) y^{d^{Ko),dl{Lo)).
Applying the functor Yl^ : (f/Ci ^(f, which preserves products since
it has a left adjoint (see 3.2.2, volume 1), we get the composite 71:
Intuitively, this composite maps a family {Ou'- K{U) ^^i^))uec ^^
morphisms to the family (Oy o K{u)) ^ , where u: U ^V. In the
same way starting with the composition morphism
7l: J^(d5(i^o), dS(i^o)) Xcx.^(dS(^o), di(io)) —^i^K(^o),dU^o))
we get the composite 70:
nc/^^)° (Ilco^" ^^^) ■ X{^f{K.,L,)-^X{^f{dlK,,d\L,).
Intuitively, this composite maps the family {Ou'- K{U) ^■^i^))uGC
to {L{u) o Ou)^^q • The object ^"^{K^L) of (pseudo)-natural
transformations from K to L is thus defined as the equalizer of the two
morphisms
3^'^{K,L)y—^^-^\[^^^{Ko,Lo)ZZ=Z:XX{^f{dl{Ko),d\{Lo)).
Let us write olkl'- ^*(i^, L) >1 for the unique morphism.
We must still define the "generic natural transformation"
fKL- a*KL{K) => a*KL{L);
here oi^l{K) and a^^iL) are just the composites
K
SI^'^{K,L)xsS{^) £-
L
where the first functor p is just the second projection. For a pair {j, m),
where
J-J ^^{K,L), m:J ^Cq,

422 Fibred categories
we must define in the fibre ^j a morphism
f{j,m)' K{J,m) >L{J,m)i
we have just dropped the indices K, L to avoid heavy notation. But
{J^m) = m*(Co,lco) ^^ S{^). Since K and L are cartesian, we must
thus define
/(,-^): m*K(Co,lco) >m*L(Co,lco),
or in other words
/(j,m): rn*(Ko) >m*(Lo).
To define it, consider the composite
J i^ J^*(i^, L) > ^ ^Hco-^^^"' ^")
in ^ = (f/I. This corresponds by adjunction with a morphism
j: J X Co >J^(Ko,^o)
in S/Cq\ thus Q^KoLo ^ J = PCo- Consider now the composite
j._(lj^ J ^ ^^ j >3^{Ko, Lo)-^^^^^Co
which is thus equal to m. It allows the definition
/(j,m) = ((lj,m)* o J*)(/koLo).
The naturality of /kl is straightforward to check, just copying the
definitions.
Finally we must check the universal property of the pair (axL? /kl)-
We consider an object M m S and the unique morphism /x: M )►!,
together with a morphism g: fi*{K) ^fJ'*{L) in the fibre over M. The
functors /Ji*{K) and fJi*{L) are just the composites
siMxsS{%)—2—>^(^):
L
and ^: K{p) =^ L(p) is a cartesian natiural transformation. We must find
a luiique morphism v\ M >^'^{K,L) such that u*{fKL) = 9-
Constructing u is equivalent to constructing 77: M >Y)lc •^(^o» -^0)
such that 7i o 77 = 70 o 77. By adjunction, constructing 77 reduces to
constructing a suitable morphism fj: M x Co >^{Ko,Lo) in ^/Cq,
i.e. aKoLo 017 = pco- To do this consider the product M xCo and its two

8.6 Locally small fibrations 423
Am X doyph,Ko=ph,dlKo "^^^ ^PhA^o = Am x dl)p}.^Ko
Am X do)*^(pM,PCo)
Am X dI)^(pM,pco)
Am X doYPc^Lo =p}.^doLo -__>p^^dJLo = Am x dJ)p^^Lo
Diagram 8.23
projections Pm^PCq- The pair (pm^PCo) is an object of ^/Mx^^(^) in
the fibre over M x Co; this yields a morphism
^(PM,Pco)- ^(^ X Co.pco) >L{M X Co,pco)
in the fibre over M x Cq. But (M x Co,pco) = Pco(^0'^<^o) i^ ^^^^
fibration ^(^); since K and L are cartesian functors, this indicates that
we have in fact obtained a morphism
)• Pcoi^o) >Pco(^o)
By universality of the pair (axoLo? /ko,Lo)> there is a unique morphism
r}: M X Co >^{Ko, Lq) such that qkoLo o ^ == PCo and TUkoLo) =
9(pM,pco)' This defines the morphism ry: M >Yl^^^{Ko^Lo).
The relation 71 o ry = 70 o ^ follows from the naturality of ^. By
adjunction, we have indeed to prove the equality of two morphisms
M X Ci ' l^{dl{Ko),dl{Lo))
Po
over Ci. By the universal construction of the codomain, it suffices to
check that
Pl(/d;(Ko),dI(Lo)) == Po{fd*{KoLl{Lo))'
Making explicit all the definitions and considering the two morphisms
1m X do, 1m X di: M X Ci Z^M x Co
one concludes that the two morphisms in the fibre of ^ over M x Ci
are the two paths in diagram 8.23. It suffices now to go back to the
definitions of k and A, namely k = K{lci)^ A = L(lci), where the
identity Ic^: (Ci,do) >{Ci,di) is a morphism of ^(^). Considering
now the product M x Ci in (f,
Am,1ci): (pM.do) >{PM,di)

424 Fibred categories
is a morphism of S'/Mx^S'{^) and the naturality of g applied to this
morphism yields the commutativity of the previous diagram. D
8.7 Definability
When F: ^ >S' is a fibration and / € ^ is an object, we think of the
objects and arrows of the fibre ^j as /-indexed families, having in mind
the canonical example Set(^) ^►Set of 8.1.9.b. As already indicated
in the introduction, there is a priori no reason for us to be able to
speak of the "subfamilies satisfying some given property". For example
if /: X >Y is a morphism in the fibre #"/, how can we indicate the
subobject J> >I in S' "where / is monomorphic" ? Being able to do it
means that the notion of monomorphism is "definable" in the fibration
F: ^ >S,
Definition 8.7.1 Let F: ^ >S' be a Gbration. A class ^ of objects
of ^ is deGnable when:
A) if a: J >I in S andX e ^i are such that X e^, then oTX € ^;
B) given objects I £ S, X ^ #"/, there exists a subobject a: J> >I
such that a*{X) € ^ and a is universal for this property, i.e. given
C: K >I such that C*{X) G ^, there exists a (unique)
factorization 7*. K >J such that /3 = a o 7.
Observe that the uniqueness of 7 is implied by the requirement that
a has to be a monomorphism.
Definition 8.7.2 Let F: ^ >^ be a Gbration. A class ^ of arrows
in the Gbres of^ is deGnable when:
A) if a: J >I in S and f: X >Y in ^i are such that / € ^, then
«*(/)€ ^;
{2) given an object I ^ S and an arrow f: X >Y in ^j, there exists
a subobject a: J> >I such that a*(/) G ^ and a is universal for
this property, i.e. given /3: K )►/ such that /3*(/) G ^, there exists
a (unique) factorization 7: K >J such that /3 = a o 7.
Definition 8.7.3 Let F: ^ >S be a Gbration. By a deGnable subG-
bration ofF we mean a subcategory^ C ^ together with the restriction
F: ^ ^<^, such that:
A) F: <« >S is still a Gbration;
B) the class of objects of^ is deGnable;
C) the class of those morphisms of^ lying in the Gbres is deGnable.

8.7 Definability 425
Proposition 8.7.4 Let j^ be a small category. The definable subfibra-
tions of the canonical fibration Set(j2/) >Set (see 8.1.9.b) are
equivalent to the canonical fibrations Set(^) ^►Set, for ^ a subcategory of
Proof li ^ C s^ is a. subcategory, Set(^) is a subfibration of Set(j2/).
Given a set / and a family {Ci)i^i of objects of ^, it suffices to put
J = {ieI\Bie^]
to satisfy condition B) of definition 8.7.1. Indeed, if C: K )►/ is such
that /3*{Ci)i^i € ^, one has C/3(fc) € ^ for each index k e K, from
which /? factors (uniquely) through the subobject J. In the same way if
ifi' Ci >Di)i^i is a family of arrows of ^, one defines
j = {iel\fie^}
to satisfy condition B) of 8.7.2.
Conversely, fix a definable subfibration ^ ^►Set of Set(j2/) ^►Set.
The fibre of Set(j2/) over the singleton 1 is precisely j^. We define ^ to
be the fibre of ^ at 1. By construction of Set(j2/), this defines ^ up to
equivalence (depending on the choice of a distinguished singleton). We
must prove that the two fibrations ^ ^Set and Set(^) ^►Set are
equivalent over Set. By 8.3.1, it suffices to prove they have equivalent
fibres.
If {Ai)i^i is an object of Set(j2/) in the fibre over /, consider the
corresponding subset J Q I given by definability of the objects of ^. For
every index i € /, we can consider the inclusion /3: {i} >I. By 8.7.1,
/3 factors through J if and only if Ai is in the fibre of ^ over {i}, i.e.
i e J a and only if Ai € ^. Thus {Ai)i^i € ^ if and only if for each
index i^ Ai € ^. The argument for the arrows is perfectly analogous. D
Counterexample 8.7.5
We consider again a category j^ and the corresponding canonical
fibration Set{rS^) ^►Set. Not all subfibrations of this fibration are definable.
For an elementary counterexample, choose j^ to be the discrete category
with just two objects X, Y. The full subcategory of Set(j2/) whose objects
are the families {Ai)i^i with all the Ai^s equal is obviously a subfibration
of Set(j2/). But it does not correspond with any subcategory of j^.
Let us now prove two important stability properties of definable
subfibrations.

426 Fibred categories
Proposition 8.7.6 If a subfibration is definable, all its localizations are
again definable subfibrations.
Proof Let F: ^ >S' be a fibration and F: ^ >S' a definable
subfibration. The locaUzed fibration F(/): ^(/) >^// is just obtained
by restricting F(/): #"(/) >^// to ^. Giving an object or an arrow in
the fibre of ^(/) over /3: K >I is just giving it in the fibre of ^ over K
(see 8.5.4). The corresponding subobject a: J) >K in S obtained by
definability of ^ immediately yields the subobject a: (J, /3a) > (K, C)
of S/I exhibiting the definability of ^(/). D
Proposition 8.7.7 If a fibration is locally small, all its definable
subfibrations are again locally small
Proof Consider a locally small fibration F: ^ >S and a definable
subfibration F\ ^ >S, Consider also I e S' and two objects X^Y
in the fibre of ^ over /. This yields by 8.6.1 two universal morphisms
axY'- ^{X,Y) >I in ^ and fxy: <^xy(^) ><^xyE^) in the fibre
of ^ over ^[X^Y). But since ^ is definable, by 8.7.2 we can find a
subobject /3xy: ^(X,y)> >^{X,Y) such that PxyUxy) is in the
fibre of ^ over ^(X, F) and is universal for this property. The composite
OLXY o PxY together with /SxyifxY) exhibits the local smallness of ^.
Indeed given 7: J >I in S' and g: 7*(X) >7*(y) in ^, we get first
a unique 6: J >^{X^Y) such that axY 0^ = 7 and 6*{fxY) = 9-
But since ^ is in ^, this morphism 6 factors through /3xy via some e,
thus /3xy o g: = 5. This yields finally a unique e: J >^{X^Y) such
that axY o /3xy o ^ == cxxy o 5 = 7 and ^*/3xy(/:^y) == ^*{fxY) = 9-
n
We conclude with some existence theorems of definable classes.
Theorem 8.7.8 Let S be a category with finite limits. If F: ^ >S'
is a locally small fibration, the class of isomorphisms is definable in ^,
Proof Consider objects I ^ S and X, F € ^i and the pullback
of diagram 8.24, where eX',SY->'yxYx->lYXY are defined in 8.6.5 and r
is the twisting isomorphism on the pullback of olxy->olyx (notation of
8.1.1).
We shall prove first that the composite
iso(x,y) ^^^ >^{x,Y)xi^{Y,x) ^^^ >j^(x,y)
is a monomorphism, which we shall write (Jxy- By definition of pull-
backs, giving an arrow p: J >lso(X, F) in S is equivalent to giving

8.1 Definability 427
\so{X,Y) ^ > I
6XY
{ex^ey)
^{X, Y)xi^{Y, X)-— r^^iX, X)xi^{Y, Y)
Diagram 8.24
arrows
u:J >^{X,Y), v:J >^{Y,X), fx: J >/,
such that
IXYZ o I \ =£x^ ^Ji', lYXY o I \ = Sy o ^.
Since cfxy^P = u^ (^xy will be a monomorphism as long as u determines
p, thus as long as u determines v and //.
By 8.1.1, u is uniquely determined by the composite /? = axy^u and
the morphism g = u*{fxY)'', in the same way v is uniquely determined
by the composite /?' = ayx o v and the morphism g^ = v*{fyx)' By
assumption on u, v,fjL,/3 = C' and moreover, with the notations of 8.6.5
( u
M = OtxX o ^x o M = (^XX o IXYX o I
= OCXYX of j = OLXY O PXY of ] = OLXY O U = C',
thus /?' = /? = //. In particular the following arrows are composable in
u*Uxy): ii*{X) >ii*{Y), v*Uyx): li*{Y) >/x*(X).
Let us prove they are mutually inverse.
By assumption diagram 8.25 is commutative. By definition of "yxYX
(see 8.6.5)
^yXYxifxx) = PyxifYx) o PxyifxY),
thus
f(^) ""JXYXjifxx)

428
Fibred categories
J
^^{X,Y)xi^{Y,X)
IXYX
ex
Diagram 8.25
¥^{X,X)
xy)
Uxy)
= [pYX ° ( ^ ) ) Uyx) o (pxY of
But by definition of ex (see 8.6.5)
(m*o^3.)(/xx) = m*Ax) = Vw
from which v''{fYx)^fJ'''{fxY) = l/x*(X)- The converse equality is proved
in an analogous way.
As a first consequence, knowledge of the morphism u determines /?' =
fji = axY o u^ but also g — v*{fYx) which is just the inverse of u*{fxY)'
So u determines both // and v and axY is a monomorphism.
We are now ready to prove that isomorphisms are definable in #".
Obviously, isomorphisms are mapped onto isomorphisms by the inverse
image functors. Next fix a morphism /: X >Y in the fibre ^/, for
some object I ^ S. The identity 1/: / >I together with the
morphism /: lj(/) >l/(^) yields a unique morphism v: I ^►#'(X,y)
such that axY o v = \i and iy*{fxY) = /• Computing the pullback of
diagram 8.26 yields a monomorphism r/ which will be proved to have
the required properties.
We have to consider a morphism //: J >I in S' such that //*(/) is an
isomorphism in the fibre ^j. Let us put u = i/o//, thus u: J >^{X^ Y)
and u*{fxY) = M*^*(/xy) = M*(/)- Considering next the morphisms
fi: J-
■^/x*(X)
>I, /x*(/)-^:/x*(F)
yields a unique factorization v: J >^(y, X) in S' such that v*(/xy) =
/^* (/) ~ ^ • Finally u^v^ji constitute a triple as at the beginning of the proof

8.7 Definability 429
lso(/)—^lso(X,F)
-T/
(^XY
I —^J^{X,Y)
Diagram 8.26
and determine a morphism w: J > lso(X, Y) such that axv^w = vo^.
This yields the required factorization J >lso(/). D
Theorem 8.7.8 and its consequences are worth a comment. A
morphism /: X >Y in a category ^ is an isomorphism when
3ge^{Y,X) fog=ly, gof = lx.
In this formula, the existential quantifier acts on a variable g running
through a set ^(F, X), thus by the comprehension scheme the formula
determines a subset lso(X, F) C ^(X, F) of those arrows /: X >Y
satisfying the required property. The local smallness of ^ has been used
twice, for ^(F, X) and for ^(X, F), for us to be allowed to apply the
comprehension scheme.
Replacing "isomorphism" by "monomorphism" would lead us to
consider the formula
VZ€^ Vix,i;€^(Z,X) fou = fov=>uov.
This formula contains a quantifier acting on a variable Z which runs
through a class ^, not just a set. Thus in most set theories, the formula
will not determine a subset Mono(X, F) C ^(X, F) of those arrows
/: X >Y which satisfy the required property, because the
comprehension scheme cannot be applied.
But now suppose the category ^ has finite limits. In that case we can
consider the kernel pair of a morphism /: X >F, as in diagram 8.27,
(see 2.5.6, volume 1). We know that / is a monomorphism if and only
if a = p^ thus if and only if the equalizer k: K >P of a,/? is an
isomorphism. This reduces the property "/ is a monomorphism" to the
properties "a is equal to /3" or "fc is an isomorphism", which can now
be handled in any set theory.
These considerations underline the fact that working with finitely
complete fibrations, in the following results, is not just a convenient technical

430
Fibred categories
a
X
-> Y
Diagram 8.27
requirement, but is really a crucial assumption. The reader could argue
that defining a finite limit requires also a quantification on the class
of all objects of the category, to express the universal property of the
limit. This is true, but this is also an observation of a different nature.
Proving a property for every object of a category does not require any
comprehension scheme at all; it is constructing the set of those objects
satisfying the property which requires the comprehension scheme.
Proposition 8.7.9 Let F: ^ >^ he a locally small and finitely
complete fibration over a category S with finite limits. The class of mono-
morphisms in the fibres of ^ is definable.
Proof Given an arrow /: X >Y in the fibre #"/ over / € ^, we
compute the equalizer kf of the kernel pair (a/,/?/) of / (see 2.5.6,
volume 1) in the fibre ^/. We know that / is a monomorphism iff
af = /3f, thus iff kf is an isomorphism.
Since inverse image functors preserve kernel pairs and equalizers, they
preserve monomorphisms. Next if /: X >Y is an arrow as above,
consider by 8.7.8 and 8.7.2 the subobject a: J) >I universal for the
fact that a*(fc/) is an isomorphism. Again since the inverse image
functors preserve kernel pairs and equalizers, A:o,*(/) = cx*{kf) and thus
a: J> >I is universal for the fact that k^^^f) is an isomorphism, i.e.
q;*(/) is a monomorphism. D
Proposition 8.7.10 Let F: ^ >^ be a locally small and finitely
complete fibration over a category S' with finite limits. The class of
terminal objects in the fibres of ^ is definable.
Proof Given an object X G ^i in the fibre over /, let us consider
the unique arrow ex'- X >1/ in ^/, where 1/ denotes the terminal
object of the fibre. Since inverse image functors preserve terminal
objects, given a morphism a: J >I in <?, a*{X) is a terminal object in

8.7 Definability 431
#'j precisely when a*{ex) is an isomorphism. Thus the universal mono-
morphism a: J> >I for a*{X) terminal is just the universal morphism
a: J> >I for a*{ex) an isomorphism (see 8.7.8). D
Definitions 8.7.1 and 8.7.2 explain the notion of definability for a class
of objects or a class of arrows. In fact the notion of definability can be
immediately extended to a class of diagrams of arbitrary shape ^.
Definition 8.7.11 Let F: ^ >S' be a Gbration and 2^ a small
category. A class ^ of diagrams of shape Si in the fibres of ^ is definable
when the class ^ of corresponding objects in the fibration #'(^) >S'
is definable (see 8.1.7).
We recall that the fibre of i^(^) over an object / G ^ is precisely
the category of diagrams of shape ^ in the fibre ^/. One observes
immediately that definition 8.7.2 is just a special case of 8.7.11, taking
for ^ the category • >• with two distinct objects and just one non-
trivial arrow between them. Here is a special case of interest:
Definition 8.7.12 Let F: ^ >& be a fibration. Equality is definable
in (#", F) when the class of parallel pairs of arrows /, g-.XZI^Y such
that f = g, in the fibres of ^^ is definable.
Proposition 8.7.13 Let F: ^ >S' be a locally small and finitely
complete fibration over a category S with finite limits. Then equality is
definable in {^,F).
Proof Given two arrows f^g: X Iv in the fibre #"/ over I e S'^
f = g precisely when the equalizer kfgi Kfg >X of {f^g) is an
isomorphism. Therefore the universal monomorphism a: J >I such that
a*{f) = a*{g) is the universal monomorphism such that a* [kfg) is an
isomorphism (see 8.7.8), just because a* preserves equalizers and thus
Proposition 8.7.14 Let F: ^ >^ be a locally small and finitely
complete fibration over a category S' with finite limits. Binary products
are definable in (#", F).
Proof We must consider the class of those pairs of arrows
x^^^—z—Py—,Y
which are the projections of the product X x F in a fibre. Another
such diagram X^f-^T—^F, is the product of X,Y when the unique
factorization t: T >Z is an isomorphism.

432 Fibred categories
Thus given a diagram X< ^ T ^ >Y in the fibre over / and the
factorization t: T >X X y as above, the universal subobject a: J> >I
such that {a*{f)^a*{g)) is a product, is just the universal subobject
such that a*{t) is an isomorphism, since a* preserves binary products.
n
More generally, one has
Proposition 8.7.15 Let F: ^ >& be a locally small and Gnitely
complete Gbration over a category S' with finite limits. Given a finite
category ^, the limits of diagrams of shape S) are definable in (^, F).
Proof Prom ^, we construct the category ^ obtained by adding one
object M to ^, an identity arrow on M and one arrow qn'- M >D
for each object of ^; we impose, for each arrow d: D >D^ in ^, the
relation do q^ = q^,, which defines the composition law in 2^. We are
interested in the class of those diagrams H: Q) ^S' j of shape Q) in the
fibres of #", for which [H{M)^ {H{qD))De^) is the limit of the diagram
H: Q) ^3^1.
For an arbitrary diagram if: Q) >^/ in the fibre over /, consider
its limit (L, (pD)r>€^) ^^d the unique factorization h\ H{M) >L such
that pDoh = H{qD) for each object D e Si. Now {H{M), {H{qD))De^)
is a limit diagram precisely when h is an isomorphism. Therefore the
universal subobject a: J> >I such that [a*H{M)^{a*H{qD))De^) is
a limit diagram is just the universal subobject such that a*{h) is an
isomorphism, because a* preserves finite limits. D
8.8 Exercises
8.8.1 Given a base category ^, construct the fibration of fibred
categories over S': the fibre at an object / € ^ is the category F\h{S'/I) of
fibred categories and cartesian functors over S/I.
8.8.2 Define (up to equivalence) the dual of a fibration F: ^ >S'
directly from (#',F), without using the axiom of choice and thus the
correspondence with pseudo-functors.
8.8.3 Let F: ^ >S' be a fibration over a finitely complete category
<?. One gets a new fibration Fam(^) >(f in the following way:
• an object of Fam(^) is a pair {a,X), where a: J >I is a mor-
phism of <? and X G ^j\

8.8 Exercises 433
• with analogous notation, a morphism {a^X) >{a\X^) is a triple
Uihf) where j: J > J', i: J > J', /: X >X^ are morphisms
and io a = a^ o j^ F{f) = i;
• the projection functor Fam(^) >S' maps {a^X) to / and {i^j^f)
to i.
Intuitively, an object of Fam(#') in the fibre over / is an /-indexed
family of families. Prove that this construction extends to a monad
Fam: Fib(^) >Fib(^). Prove that the fibration F: ^ >^ has ^-
coproducts iff the unit of the monad ^ >Fam(#') admits a fibred
left adjoint functor.
8.8.4 Choose S to be the category of commutative rings with units.
Define a fibration ^ >S by choosing as fibre over the ring R the
category of modules on R. Prove that this fibration is cocomplete.
8.8.5 Let ^ be a finitely complete and cocomplete category. Prove that
the fibration d\\ Ar(^) >S is finitely cocomplete iff finite colimits are
universal in S, When this is the case, prove that the fibration is
cocomplete.
8.8.6 Consider the canonical fibration d\\ Ar(Cat) ^►Cat of the
category Cat of small categories and functors over itself. Show that for every
category / € Cat, the "diagonal" cartesian functor Cat >Ar(Cat)^ has
a fibred right adjoint, but that nevertheless the fibration Ar(Cat) ^►Cat
is not complete, because the inverse image functors do not have a right
adjoint.
8.8.7 Let ^ be a category with finite limits. Prove that the fibration
of fibred categories over S is complete (see 8.8.1) and has ^-indexed
coproducts.
8.8.8 Let ^ be a category with finite limits, F\ 3^ >S a fibration
and Q) an internal category of S. If the fibration (#", F) is complete or
cocomplete, show that the same holds for the fibration ^^ >S,
8.8.9 Let ^ be a category with finite limits and F\ ^ >& a locally
small fibration. Given objects X € #"/, Y € ^ j in the fibres over /, J,
define an object Hom(X, y) € ^, morphisms
9o:Hom(X,y) >/, ^i: Hom(X,y) >J
in S and a morphism of ^,
(^xr: dlX >9*y.

434 Fibred categories
universal for these data. [Hint: consider the product IxJ and the object
8.8.10 Let ^ be a category with finite limits. A fibration F: ^ ^& is
locally small iff given objects X, y € ^, there exist an object Z € #", a
cartesian arrow /: Z ^X and a morphism g\ Z ^Y^ universal for
these data. [Hint: with the notation of 8.8.9, Z = d^X in the fibre over
Hom(X,y).]
8.8.11 Let ^ be a category. In the case of the fibration Set(^) >Set,
compute explicitly the objects #'(X, F), Hom(X,F), Z of 8.8.9 and
8.8.10 in the case of two objects X, Y in the same fibre.
8.8.12 Let ^ be a category with finite limits. Prove that when the
canonical fibration d\\ Ar(^) >S of S over itself has ^-indexed products, it
is locally small. [Hint: given two arrows u\ J >/, v\ K >I compute
their pullback {L,u\v^) and consider nit('^0-]
8.8.13 Let F: ^ >S' be a fibration with ^ non-empty. If the fibration
is locally small and S' has an initial object 0, prove that the fibre ^o is
equivalent to the terminal category.
8.8.14 Let F: ^ >& be a locally small fibration. Show that for every
epimorphism a: J >/ in ^, the inverse image functor a*: ^j >#'j
is faithful.
8.8.15 Let S be the category of i?-modules, for a ring R, Prove that in
the canonical fibration d\\ Ar(^) >S of S over itself, the equality is
not definable.
8.8.16 Let F\ ^ >S be a fibration and ^ a definable class of objects
of ^. Given a strong epimorphism a: J >/ and an object X € #"/,
prove that X € ^ iff q;*(X) € ^. Generalize this result by replacing a
with a strongly epimorphic family.
8.8.17 Let F\ 3^ >S be a fibration, with ^ finitely complete. If ^, Q)
are definable classes of objects in #", prove that their intersection ^fl^
is again definable.
8.8.18 Let F\ ^ ^& be a fibration for which the class of
isomorphisms in the fibres of ^ is definable. Show that when & has an initial
object 0, the fibre ^^ is a groupoid (i.e. all its arrows are invertible).
8.8.19 Let F\ ^ >8, G\ "S >S, H: Jf >S' be fibrations; let
also U: ^ >Jf, V\ ^ >Jf be cartesian functors over S. Define
a fibration (?7, V) >S^ obtained by computing fibrewise the comma
categories. If #', ^ are locally small and the equality is definable in Jf',
prove that (?7, V) is locally small.

8.8 Exercises 435
8.8.20 Suppose S' has finite limits. With the notation of 8.8.19, prove
that Jf is locally small iff for every small fibration #", ^ and all cartesian
functors C7, V, the fibration (C7, V) is small.
8.8.21 Consider fibrations F: ^ >^, G\ ^ >^, H\ ^ >^
and a cartesian functor U\ ^ >^ over S, Prove that when 3" is
small, ^ is locally small and Jf is cocomplete, the cartesian functor
Jf ^: Jf^ > Jf-^ induced by the composition with F has a fibred left
adjoint functor. [Hint: use the fibred comma categories of 8.8.19 to mimic
the proof of 3.7.2, volume 1.]

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Index
The page number 1.123 indicates page 123 in volume I
adjoint arrows, 1.304, 1.326
adjoint functors, 1.113
fibred -, 394
adjoint functor theorem, 1.124
special -, 1.125
the more general -, 1.296
adjoint lifting theorem, 226
adjunction, 1.113
algebra, 189
free -, 191
morphism of -s, 189
algebraic theory, 130
commutative -, 166
morphism of -'es, 143
presentation of an -, 125, 133
tensor product of -'es, 174, 176
arrow, 1.19
codomain of an -, 1.19
domain of an -, 1.19
identity -, 1.19
source of an -, 1.19
target of an -, 1.19
asterisk, 359
axiom, 123, 125
Banach spaces, 257
bicategory, 1.323
bifunctor, 1.38
bihomomorphism, 167
bilimit, 1.315
bimodule, 1.328, 1.329
biproduct, 5
boolean algebra, 135
^°°-algebra, 136
calculus of fractions, 1.198
cartesian closed category, 294
cartesian functor, 382
cartesian morphism, 375
cartesian natural transformation, 382
category, 1.19
n-category, 1.313
- of arrows over J, 1.22
- of arrows under J, 1.22
- of elements, 1.37
- of fractions, 1.196
- with absolute colimits, 1.291
- with enough projectives, 1.179
2-category, 1.302
3-category, 1.312
abelian -, 13, 109
accessible -, 263
additive -, 6
algebraic -, 158, 257
a-filtered -, 1.287
bicategory, 1.323
cartesian closed -, 1.298, 349
Cauchy complete -, 1.270, 266, 289
comma -, 1.35, 1.106
complete -, 1.74, 1.122
connected -, 1.73
cotensored -, 320
discrete -, 1.21
dual -, 1.48
enriched V-category, 300
exact -, 109
fibred -, 375
fibred comma -, 434
filtered -, 1.91
finitely complete -, 1.74
finitely generated -, 1.78
finitely well-complete -, 1.161
locally cartesian closed -, 1.359
locally presentable -, 1.260, 256
Mal'cev -, 121
monoidal -, 291
439

440
Index
preadditive -, 4
regular -, 1.91, 90, 91, 252
small -, 1.21
symmetric monoidal closed -, 167
tensored -, 320
well-powered -, 1.147
Cauchy completeness, 1.339
Cauchy completion, 1.353, 289
cell, 1.303
0-cell, 1.303
1-cell, 1.303
2-cell, 1.303
3-cell, 1.313
class, 1.18
colimit closed -, 1.221
saturated -, 1.206
cocone, 1.72
coend, 329
coequalizer, 1.64
split -, 212
cofibration, 382
cogenerator, 1.181
cokernel, 1
colimit, 1.72
absolute -, 1.82
a-filtered -, 1.287
filtered -, 1.91
universal -, 1.99
weighted -, 327
commutative diagram, 1.19
commutativity condition, 1.193
comonad, 219
compactly continuous, 360
compact element, 141
compact space, 236
cone, 1.71
lax-cone, 1.321
pseudo-cone, 1.321
congruence, 139
constant, 122
coproduct, 1.59
- in a fibration, 412
associativity of -, 1.60
coreflection along a functor, 1.113
cotensor, 320, 331
crossed module, 1.360
definable class, 424
- of arrows, 424
- of diagrams, 431
- of objects, 424
definable subfibration, 424
descent data, 240
descent morphism, 253
efFective -, 242
descent theory, 237
distributor, 1.329
duality principle, 1.48
abelian -, 13
Eilenberg-Moore category, 189
embedding theorem, 112
end, 329
enriched, 300
- adjunction, 340
- category, 300
- distributor, 306
- functor, 301
- Kan extension, 344
- natural transformation, 301
- Yoneda embedding, 313
- Yoneda lemma, 311
epimorphism, 1.42
extremal -, 1.151
regular -, 1.151
strong -, 1.152, 1.209
universal -, 120
equalizer, 1.63
equivalence, 1.304
equivalence of categories, 1.132
exact sequence, 32, 33, 95
short -, 34, 87
factorization, 1.162
epi-strong-mono -, 1.164
strong-epi-mono -, 1.162
factorization system, 1.225
faithful embedding theorem, 73
fibration, 375
cocomplete -,412
complete -, 402
dual -, 393
finitely complete -, 402
internally complete -, 402
localized -, 402
locally small -,412
power -, 398
small -, 379
split -, 392
fibred adjunction, 394
fibred internal limit, 402
fibred limit, 401
fibre, 374
field, 182
final object, 1.62
five lemma, 41
free abelian group, 1.117
free group, 1.117
free monoid, 1.117
free ring, 1.118
Pubini formula, 347
full and faithful embedding theorem, 80
functor, 1.20
- preserving monomorphisms, 1.39

Index
441
- reflecting monomorphisms, 1.39
- with rank, 272
2-functor, 1.308
absolutely flat -, 1.290
additive -, 8
additive representable -, 12
algebraic -, 144
a-flat -, 1.289
a-left exact -, 1.288
cartesian -, 382
collectively faithful family of -s,
1.169
contravariant -, 1.30
cotopological -, 368
covariant -, 1.31
dominated -, 183
exact -, 50, 97
faithful -, 1.33
family of -s collectively reflecting
isomorphisms, 1.169
fibre small -, 370
final -, 1.84
flat -, 274, 1.268
forgetful -, 1.23
full -, 1.33
full and faithful -, 1.34
Hom-functor, 1.49
identity -, 1.20
inverse image -, 393
lax-functor, 1.317
left adjoint -, 1.113
left exact -, 50, 1.268
limit preserving -, 1.79
limit reflecting -, 1.80
monadic -,212
pseudo-functor, 1.318
representable -, 1.24
right adjoint -, 1.113
right exact -, 50
topological -, 367
G-set, 136, 297, 298
Galois connection, 1.120
generator, 1.166
dense -, 1.167
dense family of -s, 1.167
family of-s, 1.166
strong-, 1.166,1.268, 1.171
strong family of-s, 1.166, 1.171
global support, 120
graph, 1.191
conditional -, 1.193
morphism of-, 1.191
groupoid, 1.307
group, 122
torsion free -, 232
Heyting algebra, 300
Hilbert spaces, 289
homomorphism, 126, 130
homotopy, 1.306
idempotent, 1.290
split -, 1.290
image, 19, 92
initial object, 1.62
initial structure, 366
injective object, 1.181
internal, 1.345
- base-valued functor, 1.349
- category, 1.345
- category of elements, 1.351
- colimit, 1.354
- constant functor, 1.348, 1.350
- distributor, 1.359
- dual category, 1.348
- filtered category, 1.352
- flat functor, 1.353
- functor, 1.347
- limit, 1.354
- natural transformation, 1.347, 1.350
- power, 1.356
- product, 1.356
internal automorphism, 1.307
intersection, 1.133, 26
inverse image functor, 393
isomorphism, 1.34, 1.46
Kan extension, 1.129, 1.304
fibred -, 434
pointwise -, 1.140
kernel, 2
kernel pair, 1.67
kernels lemma, 40
Kleisli category, 192
lattice, 135
V-lattice, 233
algebraic -, 141
complete -, 135, 183, 253
distributive -, 135
limit, 1.71
2-limit, 1.344, 339
a-limit, 1.287
bilimit, 1.315
fibred -, 400
fibred internal -, 401
lax-limit, 1.344, 339
pseudo-limit, 1.344, 339
weighted -, 327
localization of a category, 1.134
essential -, 1.134
localizing subcategory, 64
loop, 1.119

442
Index
Lowenheim-Skolem theorem, 278
metric space, 347
model, 124, 125, 130, 174, 277
finitely generated -, 146
finitely presentable -, 146
free -, 127, 146
modification, 1.311, 1.320
module, 135
flat -, 249
free -, 157
monad, 189
- generated by an adjunction, 193
- with finite rank, 231, 252
- with rank, 231, 276
algebra for a -, 189
free algebra for a -, 191
idempotent -, 196
left exact -, 252
morphism of -s, 229
morphism of algebras for a -, 189
- of families, 433
monoid, 186
monoidal category, 292
biclosed -, 293
morphism of -s, 313
symmetric -, 292
symmetric monoidal closed category,
293
monomorphism, 1.38
pure -, 249
more general adjoint functor theorem,
1.296
Morita equivalence, 180
Morita equivalent categories, 1.341
morphism, 1.19
bidense -, 1.253
natural transformation, 1.24, 1.31
2-natural transformation, 1.309
cartesian -, 382
constant -, 1.30
Godement product of -s, 1.28
lax-natural transformation, 1.318
object of T^—, 311
pseudo-natural transformation, 1.320
nine lemma, 42, 87
Noether isomorphism theorems, 44
first -, 44
second -, 44
normal subgroup, 3
object, 1.19
orthogonality, 1.209
partial bijections, 289
path, 1.192
- category, 1.192
pointed set, 86, 136
poset, 133, 183
precartesian morphism, 376
presentable object, 1.259, 62
product, 1.54, 1.56
- in a fibration, 411
- of categories, 1.37
associativity of, 1.55, 1.56
profunctor, 1.329
projective object, 1.177
pseudo-element, 35
pseudo-equality, 35
pseudo-image, 35
pullback, 1.65
associativity of -, 1.69
pure monomorphism, 249
pushout, 1.67
quotient, 1.147
rank, 272
reflection along a functor, 1.112
reflexive pair, 252
regular cardinal, 1.287
relation, 101, 114
category of-s, 114
composite -, 114
effective equivalence -, 102
equivalence -, 102
opposite -, 114
retraction, 1.39
retract, 1.39
section, 1.39
semi-lattice, 134
separator, 1.166
set, 1.18
small -, 1.17
sharply less, 267
short five lemma, 42
sketch, 277
model of a -, 277, 290
morphism of -es, 290
snake lemma, 48
restricted -, 45
solution set condition, 1.124, 274
space, 353
compactly generated -, 364
exponentiable -, 355
Kelley -, 364
locally compact -, 356
span, 1.326
special adjoint functor theorem, 1.125
Stone-Cech compactification, 1.128
structure-semantics adjimction, 184
subcategory, 1.34

Index
443
algebraic -, 185
epireflective -, 1.135
full -, 1.34
orthogonal -, 1.213
reflective -, 1.133
replete -, 1.133
subobject, 1.147
closed -, 1.246
closure of a -, 1.244
dense -, 1.245
suspension, 1.119
tensor, 320, 331
tensor product of Set-valued functors,
1.145
terminal object, 1.62
term, 123, 125
topology, 355
compact open -, 356
pointwise -, 299, 357
torsion free group, 110
torsion theory, 52
hereditary ~, 56
union, 1.147, 26
effective -, 27
universal algebra, 125
universal closure operation, 1.244
universe, 1.16
axiom of-s, 1.17
Yoneda embedding, 1.31, 1.32
Yoneda lemma, 1.25
additive -,12
fibred -, 385
zero morphism, 1
zero object, 1