Categories and Functors (Pure and Applied Mathematics)

CATEGORIES AND FUNCTORS

This is Volume 39 in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAULA. SMITH AND SAMUEL EILENBERG A complete list of titles in this series appears at the end of this volume

CATEGORIES AND FUNCTORS Bodo Pareigis UXIVERSITY

OF MUNICH

MUNICH, GERMANY

1970

A C A D E M I C P R E S S New York London

This is the only authorized English translation of Kuregorien und Funktoren Eine Einfirhrung (a volume in the series “Mathematische Leitfaden,” edited by Professor G. Kothe), originally in German by Verlag B. G. Teubner, Stuttgart. 1969

-

COPYRIGHT 0 1 9 7 0 , BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

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LIBRARY OF CONGRESS CATALOG CARD NUMBER: 76 - 117631

PRINTED IN THE UNITED STATES OF AMERICA

Contents Preface

1

.

.

vii

Preliminary Notions 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 1.16

2

............................

Definition of a Category . . . . . . . . . . . . . . . . . Functors and Natural Transformations . . . . . . . . . . Representable Functors . . . . . . . . . . . . . . . . . Duality . . . . . . . . . . . . . . . . . . . . . . . Monomorphisms. Epimorphisms. and Isomorphisms . . . . Subobjects and Quotient Objects . . . . . . . . . . . . . Zero Objects and Zero Morphisms . . . . . . . . . . . . Diagrams . . . . . . . . . . . . . . . . . . . . . . . Difference Kernels and Difference Cokernels . . . . . . . . Sections and Retractions . . . . . . . . . . . . . . . . Products and Coproducts . . . . . . . . . . . . . . . . Intersections and Unions . . . . . . . . . . . . . . . . Images. Coimages. and Counterimages . . . . . . . . . . Multifunctors . . . . . . . . . . . . . . . . . . . . . The Yoneda Lemma . . . . . . . . . . . . . . . . . . Categories as Classes . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . .

1 6 10 12 14 20 22 24 26 29 29 33 34 39 41 48 49

Adjoint Functors and Limits 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Adjoint Functors . . . . . . . . . . Universal Problems . . . . . . . . Monads . . . . . . . . . . . . . Reflexive Subcategories . . . . . . Limits and Colimits . . . . . . . Special Limits and Colimits . . . . Diagram Categories . . . . . . . V

.......... . . . . . . . . . . . .......... . . . . . . . . . . .

...........

. . . . . . . . . . . . . . . . . . . . . .

51 56 61 73 77 81 89

vi

CONTENTS

2.8 2.9 2.10 2.11 2.12

3

.

97 105 110 113 115 118

Universal Algebra Algebraic Theories . . . . . . . . . . . . . . . . . Algebraic Categories . . . . . . . . . . . . . . . . . Free Algebras . . . . . . . . . . . . . . . . . . . . . Algebraic Functors . . . . . . . . . . . . . . . . . . Examples of Algebraic Theories and Functors . . . . . . Algebras in Arbitrary Categories . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . .

3.1 3.2 3.3 3.4 3.5 3.6

4

Constructions with Limits . . . . . . . . . . . . . . . . The Adjoint Functor Theorem . . . . . . . . . . . . . . Generators and Cogenerators . . . . . . . . . . . . . . Special Casesof the Adjoint Functor Theorem . . . . . . . Full and Faithful Functors . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . .

. . .

120 126 130 137 145 149 156

Additive Categories . . . . . . . . . . . . . . . . . . Abelian Categories . . . . . . . . . . . . . . . . . . . Exact Sequences . . . . . . . . . . . . . . . . . . . . Isomorphism Theorems . . . . . . . . . . . . . . . . . The Jordan-Holder Theorem . . . . . . . . . . . . . . Additive Functors . . . . . . . . . . . . . . . . . . . Grothendieck Categories . . . . . . . . . . . . . . . . . . . . . . . The Krull.Remak.Schmidt.AzumayaTheorem Injective and Projective Objects and Hulls . . . . . . . . . Finitely Generated Objects . . . . . . . . . . . . . . . Module Categories . . . . . . . . . . . . . . . . . . . Semisimple and Simple Rings . . . . . . . . . . . . . . Functor Categories . . . . . . . . . . . . . . . . . . . Embedding Theorems . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . .

158 163 166 172 174 178 181 190 195 204 210 217 221 236 244

. .

. Abelian Categories 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.1 1 4.12 4.13 4.14

Appendix . Fundamentals of Set Theory

247

.........................

257

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259

Bibliography Index

. . . . . . . . . .

Thinking-is it a social function or one of the brains 7 Stanislaw Jmzy Lec

Preface

I n their paper on a “General theory of natural equivalences” Eilenberg and MacLane laid the foundation of the theory of categories and functors in 1945. It took about ten years before the time was ripe for a further development of this theory. Early in this century studies of isolated mathematical objects were predominant. During the last decades, however, interest proceeded gradually to the analysis of admissible maps between mathematical objects and to whole classes of objects. This new point of view is appropriately expressed by the theory of categories and functors. Its new language-originally called “general abstract nonsense” even by its initiators-spread into many different branches of mathematics. T h e theory of categories and functors abstracts the concepts “object” and “map” from the underlying mathematical fields, for example, from algebra or topology, to investigate which statements can be proved in such an abstract structure. Then these statements will be true in all those mathematical fields which may be expressed by means of this language. Of course, there are trends today to render the theory of categories and functors independent of other mathematical branches, which will certainly be fascinating if seen for example, in connection with the foundation of mathematics. At the moment, however, the prevailing value of this theory lies in the fact that many different mathematical fields may be interpreted as categories and that the techniques and theorems of this theory may be applied to these fields. It provides the means of comprehension of larger parts of mathematics. It often occurs that certain proofs, for example, in algebra or in topology, use “similar” methods. With this new language it is possible to express these “similarities” in exact terms. Parallel to this fact there is a unification. T h u s it will be easier for the mathematician who has command of this language to acquaint himself with the fundamentals of a new mathematical field if the fundamentals are given in a categorical language. vii

viii

PREFACE

This book is meant to be an introduction to the theory of categories and functors for the mathematician who is not yet familiar with it, as well as for the beginning graduate student who knows some first examples for an application of this theory. For this reason the first chapter has been written in great detail. The most important terms occurring in most mathematical branches in one way or another have been expressed in the language of categories. T h e reader should consider the examples-most of them from algebra or topology-as applications as well as a possible way to acquaint himself with this particular field. The second chapter mainly deals with adjoint functors and limits in a way first introduced by Kan. The third chapter shows how far universal algebra can be represented by categorical means. For this purpose we use the methods of monads (triples) and also of algebraic theories. Here you will find represented one of today’s most interesting application of category theory. The fourth chapter is devoted to abelian categories, a very important generalization of the categories of modules. Here many interesting theorems about modules are proved in this general frame. The embedding theorems at the end of this chapter make it possible to transfer many more results from module categories to arbitrary abelian categories. The appendix on set theory offers an axiomatic foundation for the set theoretic notions used in the definition of categorical notions. We use the set of axioms of Godel and Bernays. Furthermore, we give a formulation of the axiom of choice that is particularly suitable for an application to the theory of categories and functors. I hope that this book will serve well as an introduction and, moreover, enable the reader to proceed to the study of the original literature. He will find some important publications listed at the end of this book, which again include references to the original literature. Particular thanks are due to my wife Karin. Without her help in preparing the translation I would not have been able to present to English speaking readers the English version of this book.

Preliminary Notions T h e first sections of this chapter introduce the preliminary notions of category, functor, and natural transformation. T h e next sections deal mainly with notions that are essential for objects and morphisms in categories. Only the last two sections are concerned with functors and natural transformations in more detail. Here the Yoneda lemma is certainly one of the most important theorems in the theory of categories and functors. T h e examples given in Section 1.1 will be partly continued, so that at the end of this chapter-for some categories-all notions introduced will be known in their specific form for particular categories. T h e verification that the given objects or morphisms in the respective categories have the properties claimed will be left partly to the reader. Many examples, however, will be computed in detail.

1.1 Definition of a Category I n addition to mathematical objects modern mathematics investigates more and more the admissible maps defined between them. One familiar example is given by sets. Besides the sets, which form the mathematical objects in set theory, the set maps are very important. Much information about a set is available if only the maps into this set from all other sets are known. For example, the set containing only one element can be characterized by the fact that, from every other set, there is exactly one map into this set. Let us first summarize in a definition those properties of mathematical objects and admissible maps which appear in all known applications. As a basis, we take set theory as presented in the Appendix. Let V be a class of objects A , B, C ,... E Ob V together with (1)

A family of mutually disjoint sets {Mor,(A, B)} for all objects A, B E V, whose elements f, g, h ,... E Mor,(A, B ) are called morphisms and 1

2

1. PRELIMINARY NOTIONS

(2) a family of maps {Morop(A,B ) x MordB, C ) 3 (f,g) - g f c MordA, C ) ) for all A, B, C E Ob V, called compositions. V is called a category if V fulfills the following axioms: (1) Associativity: For all A , B, C, D E Ob V and allfE MorlB(A,B), g E Mor,(B, C ) , and h E Moro(C, D)we have

4gf) = (k)f ( 2 ) Identity: For each object A E Ob V there is a morphism 1, E Mory(A, A), called the identity, such that we have fl, = f

and

l,g = g

for all B, C E Ob V and all f E Mory(A, B) and g

E

Mory(C, A).

Therefore the class of objects, and the class of morphism sets, as well as the composition of morphisms, always belong to a category V. T h e compositions have not yet been discussed in our example of sets, whereas the morphisms correspond to the discussed maps. I n the case of sets the composition of morphisms corresponds to the juxtaposition of set maps. This juxtaposition is known to be associative. T h e identity map of a set complies with the axiom of identity. Thus all sets together with the set maps and juxtaposition form a category, which will be denoted by S. Here it becomes clear why one has to consider a class of objects. I n fact because of the well-known inconsistencies of classical set theory, the totality of all sets does not itself form a set. One of the known ways out of this difficulty is the introduction of new boundless sets under the name classes. This set theory will be axiomatically treated in the Appendix. A further possibility is to ask axiomatically for the existence of universes where all set theoretic constructions do not exceed a certain cardinal. In some cases this makes possible an elegant formulation of the theorems on categories. It requires, however, a further axiom for set theory. This possibility was essentially used by A. Grothendieck and P. Gabriel. W. Lawvere developed a theory in which categories are axiomatically introduced without using a set theory and from which set theory is derived. Here we shall only use the set theory of Goedel-Bernays (Appendix). Before examining further examples on categories, we will agree on a sequence of abbreviations. In general, objects will be denoted by capital Latin letters and morphisms by small Latin letters. T h e fact that A is

1.1

3

DEFINITION OF A CATEGORY

an object of % will be expressed by A E V, and f E V means that f is a morphism between two objects in W , that is, there are two uniquely defined objects A, B E % such that f E Mor,(A, B). A is called the domain off and B the range off. We also write f:A+B

or

A L B

If there is no ambiguity, Mor,(A, B) will be abbreviated by Mor(A, B). Mor % denotes the union of the family of morphism sets of a category. Observe that Mor,(A, B) may be empty, but that Mor V contains at least the identities for all objects so that it is empty only for an empty class of objects. Such a category is called an empty category. Observe further that for each object A E W ,there is exactly one identity 1, . If lA‘is another identity for A then we have I,’ = l,’l, = 1, . In the following examples only the objects and morphisms of a category will be given. The composition of morphisms will be given only if it is not the juxtaposition of maps. We leave it to the reader to verify the axioms of categories in the following examples. Examples 1. S-Category Appendix.

of sets: This is sufficiently described above and in the

2. Category of ordered sets: An ordered set is a set together with a relation on this set which is reflexive (a E A =- a < a), transitive (a < b, b < c * a < c), and antisymmetric (a < b, b \< a => a = b). The ordered sets form the objects of this category. A map f between two ordered sets is order preserving if a ,< b implies f (a) f (b). The order preserving maps form the morphisms of this category.


>= d C > ( d a ) > ( f )

hence, v(A)(a)Mor(g, B) = v(C)g(a). So the diagram is commutative. In linear algebra one finds a corresponding natural transformation from a vector space to its double dual space.

1.4

Duality

We already noticed for contravariant functors that they exchange the composition of morphisms in a peculiar way, or, expressed in the language of diagrams, that the direction of the arrows is reversed after the application of a contravariant functor. This remark will be used for the construction of an important functor. Let us start with an arbitrary category V. From V we construct another category V owhose class of objects is the class of objects of V whose morphisms are defined by Morw(A, B) := Mor,(B, A), and whose compositions are defined by Moryo(A, B ) x Morw(B, C ) 3 (f,g) t+fg

E

Mor,rJ(A, C )

with fg to be formed in V. It is easy to verify that this composition in Vo is associative and that the identities of V are also the identities in Vo. The category V ois called the dual category of V. The applications V 3 A t+ A E Vo Moryp(A,B) 3f H f E Mor,,(B, A)

and Vo3 A H A

EV

Morw(A, B ) 3 f - f ~ Mory(B, A)

define two contravariant functors, the composition of which is the identity on V and Vorespectively. T o denote that A [ f ]is considered as an object [a morphism] of Vo we often write Ao[fO]instead of A [orf]. By definition, we have for every category V = (V0)O. The functors described here will be Iabeled by Op: V + 'ifo and Op: Vo---t 'if respectively. Both functors exchange the direction of the morphisms or, in

1.4

13

DIJALITY

diagrams, the direction of the arrows and thereby simultaneously the order of the composition, no other composition for categories being defined. In fact we have f o g o = (gf)O. If we apply this process twice, we get the identity again. From this point of view, the second part of the lemma in Section 1.3 could be proved in the following way. Instead of examining the maps S , we examine the maps Morqo(A, -): defined by Mor,(-, A ) : V V o--f S . By the first part of the lemma, these maps form a functor. I t is easy to verify that Morvo(A, -) Op = MOT,(-, A ) considered as maps from V to S. Consequently, Mor,( -, A ) is a contravariant functor. Instead of proving the assertion for V , we proved the “dual assertion” for go,the dual assertion being the assertion with the direction of the morphisms reversed. Thus, to each assertion about a category, we get a dual assertion. An assertion is true in a category ‘2?if and only if the dual assertion is true in the category Yo. We want to describe this so-called duality principle in a more exact way with the set theory presented in the Appendix. Let g(V) be a formula with a free class variable V . 5 = 5(V) is called a theorem on categories if --f

( A ?7)(%? is a category +- g(V))

is true, that is, if the assertion g(W) is true for all categories V . From 8 we derive a new formula 5 O = go(5a) with a free class variable 9 by

go(9) = ( V U)(U is a category A VO

=9 A

5(?7))

that is, gO(9) is true for a category 9 if and only if g(9O) is true because Vo = 9 implies %? = 9 0 . If g(V) is a theorem on categories, we get go(%) from g(V) by reversing the directions of all morphisms appearing in g(V). This corresponds exactly to the construction of g(Vo).go is called the dual formula to 5. T h u s we get the following duality principle: Let 8 be a theorem on categories. Then go, the dual formula to is also a theorem on categories, the so-called dual theorem to 3.

5,

I n fact, if g(V) is true for all categories G f , then %(go) is true for all categories V and consequently also g(’(W). When we apply this duality principle, we have to bear in mind that we dualize not only the claims of the theorems on categories but also the hypotheses. When we introduce new abbreviating notions, we have to define the corresponding dual notions also.

14

1.

PRELIMINARY N O T I O N S

1.5 Monomorphisms, Epimorphisms, and Isomorphisms In the theory of categories, one tries to generalize as many notions as possible from special categories, for example the category of sets, to arbitrary categories. An appropriate means of comparison with S are the morphism sets, or more precisely, the covariant representable functors from an arbitrary category U into S. So the property E could be assigned to an object A E U [a morphism f E U ] if A [ f ] is mapped by each representable functor Mor,(B, -) to a set [a map] in S with the property (3. In order to recover the original definition in the case U = S, we have to observe further that the property E of a set or map is preserved by Mor,(B, -) and is characterized by this condition. We find a first application of this principle with the notion of an injective set map. Let f : C -+ D be an injective map. Then Mor(B,f): Mor(B, C) --f Mor(B, D)is injective for all B E S. In fact, Mor(B,f)(g) = Mor(B,f)(h) for all g, h E Mor(B, C) implies f g = f h . So we have f ( g ( b ) ) = f ( h ( b ) )for all b E B. Since f is injective, g ( b ) = h(b) for all b E B, that is g = h. Consequently, it makes sense to generalize this notion because the converse follows trivially from B = { 0}. Let V be a category and f a morphism in V. f is called a monomorphism if the map Mor,(B, f ) is injective for all B E V. We define the epimorphism dual to the notion of the monomorphism. Let $f be a category and f a morphism in V. f is called an epimorphism if the map Mor,( f , B) is injective for all B E V.

LEMMA 1. (a) f : A

--t B is a monomorphism in V i f and only if f g = fh implies g = h for all C E V and for all g , h E Mory(C, A), that is, i f f is

left cancellable. (b) f : A -+ B is an epimorphism in V i f and only i f gf = hf implies g = h for all C E V and for a l l g , h E Mor,(B, C), that is, i f f is right cancellable. Proof. (a) and (b) are valid because Mor(C,f)(g) = f g and Mor(f3 C ) ( g ) = gf. T h e following two examples show that monomorphisms [epimorphisms] are not always injective [surjective] maps if the morphisms of the category in view can be considered as set maps at all. Examples

1. An abelian group G is called divisible if nG = G for each natural number n, that is, if for each g E G and n there is a g' E G with ng' = g. Let V be the category of divisible abelian groups and group homomor-

1.5

MONOMORPHISMS, EPIMORPHISMS, AND ISOMORPHISMS

15

phisms. T h e residue class homomorphism v : P -+P/Z from the rational numbers to the rational numbers modulo the integers is a monomorphism in the category V, for if f , g : A + P are two morphisms in 3 ' with f # g, then there is an a E A with f ( a ) - g(a) = YS-l # 0 and s # f l . Let b E A with rb = a. Then r ( f ( b ) - g(b)) = f ( a ) - g(a) = rs-l, s o f ( b ) = g(b) = s-l. Therefore vf(b) # vg(b). Thus, v is a monomorphism which is not injective as a set map.

2. In the category Ri epimorphisms are not necessarily surjective. T h e embedding A : Z + PI for example, is an epimorphism. Let g, h : P -+ A be given with gX = hA. Then g(n) = h(n) for all natural numbers n and g( 1) = h( 1) = 1. Hence g(n)g( 1In) = 1 = h(n) h( 1In). Thus we get g ( l / n ) : (g(n))-l = (h(n))-l = h ( l / n ) and more generally g(p) = h(p) for all p E P, that is, A is an epimorphism.

3. We give a third topological example. A topological space A is called hausdorff if for any two distinct points a, b E A there are two open setsU and V with U E U C A and b E V < 5 A such that U n V = O . T h e hausdorff topological spaces together with the continuous maps form a subcategory Hd of Top. A continuous map f : A + B is called dense if for every open set U f- o in B, there is an a E A withf(a) E U . T h e embedding P -+ R, for example, is a dense continuous map. We show that each dense continuous map in Hcl is an epimorphism. Letf : A + B be such a map. Given g :I3 + C and h : B --+ C in Hd with g # h such that g(b) # h(b) for some b E B. Then there are open sets U and V with g(b) E U C C and h(b) E V C C and U n V = o . T h e sets g-1( U ) C B and h-l( V )C B are open sets with g-l( U ) n kl( V )3 b, g and h being continuous. Furthermore, g-l( U ) n h-l( V )is a nonempty open set so that there is an a E A witlif(a) Eg-'( U ) n h-l( V). But then gf(a) E U and h f ( a ) E V . U n V = o implies g f ( a ) # hf(a), that is, gf # hf. P and R being hausdorff spaces the embedding P + R is an example of an epimorphism which is not surjective as a set map. COROLLARY (cube lemma). Let jive of the six sides of the cube A,

:~

A,

A,

+

except the top be commutative and let '4, the top side is also commutative.

-+

A,

A, be a monomorphism. Then

1.

16

PRELIMINARY NOTIONS

Proof. All morphisms in the diagram from A, to A, are equal, in particular A, -+ A, -+ A4-+ A,

and

A,-+ A2+ A, -+ A , ,

Since A, -+ A, is a monomorphism, the top side is commutative.

LEMMA 2. Let f and g be morphisms in a category which may be composed. Then: (a) (b) (c) (d)

If f g is a monomorphism, then g is a monomorphism. I f f andg are monomorphisms, then f g is a monomorphism. If fg is an epimorphism, then f is an epimorphism. I f f and g are epimorphisms, then f g is an epimorphism.

Proof. The assertions (c) and (d) being dual to the assertions (a) and (b), it is sufficient to prove (a) and (b). Let gh = gk, then fgh = fgk and h = k . This proves (a). (b) is trivial if we note that monomorphisms are exactly the left-cancellable morphisms. Example Now we want to give an example of a category where the epimorphisms are exactly the surjective maps, namely the category of finite groups. The same proof works also for the category Gr. First, each surjective map in this category is left cancellable as a set map and consequently as a group homomorphism. So we have to show that each epimorphism f : G' -+ G is surjective. We have to show that the subgroupf(G') = H of G coincides with G. Since f can be decomposed into G' -+ H -+ G, the injective map H 4 G is an epimorphism [Lemma 2(c)]. We have to show the surjectivity of this map. Let G / H be the set of left residue classes g H with g E G. Furthermore, let Perm(G/H u { 03)) be the group of permutations of the union of G/H with a disjoint set of one element. This group is also finite. Let u be the permutation which exchanges H E G / H and co, and leaves fixed all other elements. Then u2 = id, Let t : G -+ Perm(G/H u (03)) be the map defined by t(g)(g'H) = gg'H and t(g)(m) = 03. Then t is a group homomorphism. Let s : G + Perm(G/H u {a}) be defined by s(g) = ut(g)u. Then s is also a group homomorphism. One verifies elementwise that t(h) = s(h) for all h E H. Since H -+ G is an epimorphism, we get t = s. So for all g E G, gH = t ( g ) ( H ) = s(g)(H) = ut(g) u ( H ) = ut(g)(w)= ~ ( 0 0 )= H

This proves H = G.

1.5

MONOMORPHISMS, EPIMORPHISMS, AND ISOMORPHISMS

17

Let V be again an arbitrary category. A morphism f E Morw(A, B ) is called an isomorphism if there is a morphism g E Morw(B,A) such that f g = 1, and gf = 1, . Two objects A, B E V are called isomorphic if Mor,(A, B ) contains an isomorphism. Two morphisms f : A -+ B and g : A’ + B’ are called isomorphic if there are isomorphisms h : A -+ A’ and K : B + B’ such that the diagram

A-LB

hlA’ -. B’Ik g

is commutative. T h e following assertions are immediately clear. I f f : A -+ B is an isomorphism with f g = 1, and gf = l A, then g is also an isomorphism. We write f -l instead of g because g is uniquely determined by f . T h e composition of two isomorphisms is again an isomorphism. T h e identities are isomorphisms. So the relation between objects to be isomorphic is an equivalence relation. Similarly, the relation between morphisms to be isomorphic is an equivalence relation. Isomorphic objects and morphisms B and f g respectively. Now let 9 : V + 9 be are denoted by A a functor and f E V an isomorphism with the inverse isomorphism f -l. Then 9( f ) 9( f -I) = 9 ( f l - l ) == 9 ( 1 ) = 1 and analogously 9 ( f - 1 ) 9 ( f ) = 1. So the fact that f is an isomorphism implies that 9 f is also an isomorphism. A morphism f E Mor,(A, A ) whose domain and range is the same object is called an endomorphism. Endomorphisms which are also isomorphisms are called automorphisms.

LEMMA 3. I f f is an isomorphism, then f is a monomorphism and an epimorphism. Proof. Since there is an inverse morphism for f, we get that f is left and right cancellable. Note that the converse of this lemma is not true. We saw, for example, that X : Z + P in Ri is an epimorphisnn. Since this morphism is injective as a map and since all morphisms in Ri are maps, h is also left cancellable and consequently a monomorphism. h is obviously not an isomorphism because otherwise X would have to remain an isomorphism after the application of the forgetful functor into’ S, so h would have to be bijective. Similarly, v : P ---t P/Z is a monomorphism and an epimorphism in the

18

1.

PRELIMINARY NOTIONS

category of divisible abelian groups, but not an isomorphism. T h e same is true in our example of the category of hausdorff topological spaces. A category V is called balanced if each morphism which is a monomorphism and an epimorphism is an isomorphism. Examples are s, Gr, Ab, and .Mod. Let CJJ : F + 9 be a natural transformation of functors from V to 9, CJJ is called a natural isomorphism if there is a natural transformation t,4 : 9 + F such that $CJJ = idF and CJJ# = id, . Two functors F and 9 are called isomorphic if there is a natural transformation between them. Then we write F g 9.Two categories are called isomorphic if there are functors F : V + 9 and 9 : 9 + V such that 9 9 = Idv and 9 9 = Id9 . Two categories are called equivalent if there are functors F : V + 9 and 9 : 9 + V such that 9 9 gz Idw and 9 9 g Id, . T h e functors 9 and 9 are called equivalences in this case. If 9 and 9 are contravariant, one often says that V and 9 are dual to each other. If CJJ is a natural isomorphism with the inverse natural transformation 4, then $ is also a natural isomorphism and is uniquely determined by CJJ. y is a natural isomorphism if and only if cp is a natural transformation and if CJJ(A) is an isomorphism for all A E V. In fact the family {(cJJ(A))-~} for all A E V is again a natural transformation. We have to distinguish strictly between equivalent and isomorphic categories. If V and 9 are isomorphic, then there is a one-one correspondence between Ob V and Ob 9. If V and 9 are only equivalent, then we have only a one-one correspondence between the isomorphism classes of objects of V and 9 respectively. I t may happen that the isomorphism classes of objects in V are very large, possibly even proper classes, whereas the isomorphism classes of objects in 9 consist only of one element each. It is even possible to construct for each category V an equivalent category 9 with this property. I n order to do this, we use the axiom of choice in the formulation given in the Appendix. T h e notion of isomorphism defines an equivalence relation on the class of objects of V. Let Ob 9 be a complete set of representatives for this equivalence relation. We complete Ob 9 to a category 9 by defining Mora(A, B) = Mor,(A, B) and by using the same composition of morphisms as in V. Obviously 9 becomes a category. Let F : % + 9 assign to each A E V the corresponding representative F A of the isomorphism class of A. Let 21 be the isomorphism class ofA and CJthe class of those isomorphisms which exist between the elements of 2l with range F A . Let two isomorphisms be equivalent if their domain is the same. Then a complete set of representatives defines exactly one isomorphism between each element of 2l and F A . This can be done simultaneously in all isomorphism classes of objects of V. Now let f : A + B be a morphism in V.

1.5

MONOMORPHISMS, EPIMORPHISMS, AND ISOMORPHISMS

Then we assign to f the morphism SJf:S A -+ S A LB

.FA

19

B defined by

9~

Because of the commutativity of

9 is a functor from V to 9. 9 being a subcategory of V we define g : 9 -+ %' as the forgetful functor'. Trivially S g = Id,. O n the other hand, S 9 A = . F A A for all A E V . T h e diagram F9 f

%%A -+

dl A-

.F%B

?I1 f - + B

is commutative for all morphisms f E V. T h u s V is equivalent to 9.We call the category 59 a skeleton of V. Observe that by our definition the Idual category V oof V is dual to V , but that, conversely, the condition that 9 is dual to V implies only that 9 is equivalent to V0.In this context we also want to mention how contravariant functors may be replaced by covariant functors. T h u s it suffices to prove theorems only for covariant functors. As we saw, the isomorphism Op : %' -+ V o (because of the contravariance of O p this is also called antiisomorphism) has the property OpOp = Id. If 9 : V -+ 9 is a contravariant functor, then 9 O p : V0 -+ 2 and O p S : 9 -+ go are covariant functors, which may ;again be transformed into 9 by an additional composition with Op. If cFand 9 are contravariant functors from V to 9 and if q~ : F -+ 3 ' is a natural transformation, then we get corresponding natural transformations TOP : S o p -+ S o p and Opp, : O p g -+ O p 9 , as is easily verified. Let V be a small category, and let us denote the category of contravariant fuinctors from V to 9 by Functo(V,9), then the described applications between co- and contravariant functors define isomorphisms of categories

We leave the verification of the particular properties to the reader. I n particular, we get Funct(V, 9)E Funct(Vo,

20

1.

PRELIMINARY NOTIONS

1.6 Subobjects and Quotient Objects Let V be a category. Let W be the class of monomorphisms of V. We define an equivalence relation on 9N . by the following condition. Two monomorphisms f : A + B and g : C -+D are equivalent if B = D and if there are two morphisms h : A + C and k : C + A such that the diagrams A

C

C

A

are commutative. Obviously this is an equivalence relation on YJl. Let U be a complete set of representatives for this equivalence relation. U exists by the axiom of choice. Let f and g be equivalent. Then f = gh and g = fk, hence f 1, = f = fkh andgl, = g = ghk. Since f a n d g are left cancellable, we get 1, = kh and lc = hk, thus A E C. Let B E % . A subobject of B is a monomorphism in U with range B. A subobject f of B is said to be smaller than a subobject g of B if there is a morphism h IGV such that f = gh. By Section 1.5, Lemma 2(a) and since g is cancellable, h is a uniquely determined monomorphism.

LEMMA1. The subobjects of an object B E V form an ordered class.


C

h, exists because (gf)-l(C,) is a counterimage. h, exists because g-'(C,) is a counterimage. Finally, h, exists because f-l(g-l( C,)) is a counter-

1.13

IMAGES, COIMAGES, AND COUNTERIMAGES

37

image of g-l(C,). T h e monomorphisms from (gf)-l(C,) and from f-l(g--l(C,)) into A are equivalent, thus the corresponding subobjects are equal. (g) We start with the commutative diagram

A-B-

f

>c

&?

h is a monomorphism because (gf)(Al) and g ( f ( A , ) ) are subobjects of C. h is an epimorphism becausef(A,) and g ( f ( A , ) ) are epimorphic images. Thus h is an isomorphism, since % is balanced.

--

(h) We have the commutative diagram A1

1

1 A-

fV1)

f

B

f ( A , ) fulfills the property of an image for A, . Consequently, it fullfills this property also forf-'(A,). (i) is proved similarly to (h). (j) We start with the commutative diagram

38

1.

PRELIMINARY NOTIONS

f(u

We want to prove that A,) is the union of f(Ai).Let there be a morphism hli for each i E I . Because of the property of a counterimage of g-l(C,), there is a morphism h,, for all i. Then h, exists because A i is a union. h, exists becausef(U Ai) is an image. Thus we have a morphismf(u A,) + C , , fulfilling the conditions of a union.

u

(k) We start with the commutative diagram

n

is an intersection. h, exists uniquely such h, exists uniquely because that the diagram becomes commutative, becausef-l(n B,) is a counterimage. Thus thef-l(n B,) is the intersection of thef-l(Bi). We give some examples of categories satisfying all conditions of this theorem. However, we shall not verify these conditions, since they are implied by later investigations. T h e categories S, S*,Gr, Ab, ,Mod, Top, Top*, and Ri have epimorphic images, monomorphic coimages, counterimages, intersections, and unions. Except for Top, Top*, and Ri, they are all balanced.

LEMMA 3. Let V be a category with epimorphic images. V is balanced if and only if V has monomorphic coimages and if these coimages coincide up to an isomorphism with the images of the corresponding morphisms. Proof. Let V be balanced. Let ( A L B )= ( A A I m ( f ) L B )

=(

A-GCLB)

with an epimorphism h. We split h‘ in (C Im(h’) .% B). Then k’ is a monomorphism, through which f may be factored. Thus, there is a morphismf’ : Im( f )-+Im(h’) withg’ = k‘f’. Sincef = g‘g = k’f’g = k’kh, we also havef’g = kh, for k’ is a monomorphism. Since kh is an epimorphism,f’is an epimorphism. Furthermore, f ’is a monomorphism, because g’ is a monomorphism. Since V is balanced, f ’ is an isomorphism with inverse morphismf*. Thus, g = f*kh, that is, the quotient object of A , equivalent to Im( f ), is a coimage off, and the corresponding morphism into B is a monomorphism.

1.14

39

MULTIFUNCTORS

Conversely, let V be a category with monomorphic coimages which coincide up to an isomorphism with the images, and let f : A + B be a monomorphism and an epimorphism. Then A is an image off up to an isomorphism, and B is a coimage off up to an isomorphism. Thus, f is an isomorphism.

1.14 Multifunctors After having investigated the essential properties of objects and morphisms, we now have to deal with functors and natural transformations. First, let us take three categories a‘,SY, and V. T h e product category d x 9?is defined by O b ( d x SY) == O b ( d ) x O b ( a ) and

Correspondingly, we define and the compositions induced by d’ and 9. the product of n categories. It is easy to verify the axioms for a category. A functor from a product category of two [n] categories into a category V is called bifunctor [multifunctor].Special bifunctors Pd : d x SY + d are defined by Y d ( A ,B) = A and Pd(f,g) = f, and correspondingly for P a . They are called projection functors. For n-fold products, they are defined correspondingly.

LEMMA 1. Let SB: d‘ + %? and gd4 : SY -+V be functors for all A E a? and B E 93.If we have

-

for all A, A’ E d,B, B’ E 93 and all morphisms f : A + A’, g : B B’, then there is exactly one bifunctor . 9 : d‘ x SY + V with %(A, B)= g,4(B)and x (f, g) = cFl?’( f ) g,4(g)* Proof. We define 2 by the conditions for 3Y given in the lemma. T h e n one checks at once that X (1, , I B ) = l H ( a , B ) and 3Y(f ’f,g’g) = Z(f ’ 9 g’) Wf, g).

If a bifunctor If : d’x SY -+ $? is given, then .FB(A)= #(A, B) and FB( f ) = If(f, lB)is a functor from d’ into V, and correspondingly, we can define a functor g A from 28 into V. For these functors, the equations of Lemma 1 are satisfied.

1.

40

PRELIMINARY NOTIONS

COROLLARY. Let 2 and &' be bifunctors from a2 x 5$?into V. A family of morphisms v(A,B ) :*(A, B ) -+&"(A, B ) ,

A ~ d B ,E 99

is a natural tranformation if and only i f it is a natural transformation in each variable, that is, i f q ~ ( - ,B ) and ?(A, -) are natural transformations.

B) instead of %'( f , I,), Proof. If we write 2(f, following commutative diagram a(A, B )

*(A, B )

*(A', B )

m(s.8)

then we get the

'#'(A, B )

44' B )

f l ( A ' ,B)

#'(S,g)

a(A',B')

*(A', B')

LEMMA 2. For each category Mor,(-,

&"(A', B')

-) : V o x V --t S is a bifunctor.

Proof. In the lemma of Section 1.3, we proved that Mor,(A, -) : V -+S and MOTy(:, B) : Y o-+ S are covariant functors. Furthermore, because of the associativity of the composition of morphisms, we have Mor&

B') Mor*(A, g)

=

Mory(A', g) Mor@(f,B ) = : Morv(f, g)

I n particular, we have Mar,( f,g)(h) = ghf, if the right side is defined. Thus by Lemma 1, Mor,( -, -) is a bifunctor.

If we do not pass over the dual category Y o in the first argument of Mor,(--, -), then Mor,(-, -) is contravariant in the first argument and covariant in the second argument. We denote the representable functor Mor,(A, -) by hA and the representable functor Mor,(-, B) by h , . Because of the commutativity Mordf, B') MorV(A, g) = Morw(A', g) Moryp(f, B )

we have natural transformations Mor,(f, -) : Mory(A, -)

-+

Mor,(A',

-)

and MOT,( -, g) : Morg( -, B ) -+Mary( -, B')

1.15

THE YONEDA LEMMA

41

We denote Morw(f, -) by hf and Mary(-,g) by h, . These considerations lead to the following lemma.

LEMMA 3. Let d ,A$ be small categories and %? be an arbitrary category. Then we have Funct(d x A9,59)

Funct(d, Funct(A9, 59)) E Funct(A9, Funct(d, %‘))

Proof. Obviously &‘x 9 g a x d.T h u s it suffices to prove the first isomorphism. If one transfers the considerations on natural transformations made above to the general case of a bifunctor, then the application for the functors is described by Lemma 1 . T h e natural transformations are transferred in accordance with the corollary. For the applications described above, it is easy to verify the properties of a functor and the reversibility.

1.15 The Yoneda Lemma In this section we want to discuss one of the most important observations on categories. Several times we shall meet set-theoretic difficulties of the kind that one wants to collect proper classes to a set which is not admissible according to the axioms of set theory (see Appendix). Since these classes are not disjoint, we cannot even fall back on a system of representatives. This is true in particular for the natural transformations between two functors 9 : %? + 9 and 8 : %‘-+ 9. We agree on the following abbreviation: for ‘‘q : 9-+ 8 is a natural transformation” we also write “g, E M o r f ( F , 8)”or “Morl(F, 8)3 g,.” Here we do not think of Mort(*, 8)as of a set or class. If $? is a small category, however, then the natural transformations from 3 to 9 form a set, denoted by Mor,(F, 8), by the considerations of Section 1.2. In this case, the abbreviation introduced above has the further meaning ‘‘vis an element of the set Mort(B, 8).”T h e condition that %? is a small category prevents these set theoretic difficulties. Also, for further constructions, we shall generalize the usual notation, and we shall explain in each case the meaning which we attribute to the notation. T h e notation (‘7

: Mort(*,

9)3 y ++x E X”

shall mean that to each natural transformation from 9into 8 there is an element in X , a set or a class, uniquely determined by an instruction explicitly given and denoted by r . We assign a corresponding meaning to “U : X 3 x + g, E Mort($, 9).” By “Mor,(F, ’3) X” we mean that

42

1.

PRELIMINARY NOTIONS

the application T is unique and invertible. With these conventions we can carry on the following considerations as if V were a small category.

THEOREM (Yoneda lemma). Let % be a category. Let 9 : 5F? a covariant functor, and A E C be an object. Then the application T

: Morf(hA,9) 3 v H v(A)(1")

ES

--+

S be

(A)

is unique and invertible. The inverse of this application is 7-1

: S ( A )3 a ++

ha E Morf(hA,9)

where h a ( B ) ( f ) = 9 ( f ) ( a ) . Proof. If one notes that p(A) : hA(A)= Mor,(A, A) + 9 ( A ) ,then it is clear that T is uniquely defined. For +, we have to check that ha is a natural transformation. Later on we shall discuss the connection with the symbol hf,defined for representable functors Given f : B --f C in V. Then the diagram Mory(A, B ) h'(W

1

9(B)

Mor(A,f)

Mory(A, C ) IhYC)

F(f)

9(C)

is commutative, for ha(C)Mor(A,f)(g) = h"(C)(fg) = 9( fg)(a) = E Mor,(A, B). T h u s 7-l is uniquely defined. Let p = ha. Then hU(A)(1,)= 9 ( l A ) ( a )= a. Let a = p(A)(lA). Then h"(B)(f 1 = F ( f)(a) = P ( f)(P(41,4)) = v ( B )Mor(A,f)(Ll) = t.p(B)(f ), thus h" = p. This proves the theorem.

9( f ) F ( g ) ( a )= S(f ) ha(B)(a) for all g

Let 9 = hC be a representable functor. Then for f ~ 9 ( A=) hC(A)= Mor(C, A) we have the equation h'(B)(g)

=

= fg =

M o U , B)(g)

that is, the definition for hf given in the Yoneda lemma coincides in the special case of a representable functor 9 with the definition in Section 1.14. Now we want to investigate what happens with the application 7 if we change the functor Fand the representable functor hA.T h e commutative diagrams used in the following lemma are to be interpreted in such a way that the given applications coincide.

1.15

THE YONEDA LEMMA

43

LEMMA1. Let 9 and % be functors from V into S , and let tp : 9+ $9 be a natural tranformation. Let f : A + B be a morphism in V. Then the following diagrams are commutativt!:

COROLLARY 1. Let V be a small category. Then Mor,(h-, -) : %? x Funct(V, S) -+ S

and

@ : V x Funct(V, S) -+ S

are bifunctors. The application T is a na;!uralisomorphism of these bifunctors. Proof, This assertion follows from the preceeding one and from Section 1.14. T h e functor in Corollary 1 denoted by @ will be called the evaluation functor. Now we want to apply the new results for representable functors.

44

1.

PRELIMINARY NOTIONS

COROLLARY 2. Let A, B E V. Then: (a) Mor,(A, B) 3f tt hf E Morf(hB,hA) is a bijection. (b) The bijection of (a) induces a bijection between the isomorphisms in Mor,(A, B ) and the natural isomorphisms in Mor,(hB, hA). (c) For contravariant functors S : V ---t S, we have Morf(h, , S)g S(A). (d) Mor,(A, B) 3f tt h, E Mor,(h, , h,) is a bijection, inducing a bijection between the isomorphisms in Mor,(A, B ) and the natural isomorphisms in Mor,(hA , h,).

Proof. (a) is the assertion of the Yoneda Lemma for S = hA. (c) and ( d ) arise from dualization. (b) By hfhg = hgf, isomorphisms are carried over the natural isomorphisms. Conversely, let hf : hB+ hA and hg : hA + hB be inverse natural isomorphisms. Then hgf = idhA and hfg = id,B . We also have hlA = idhA and hlB = idhB ,thusgf = 1, and f g = l B. The properties of h we used in the preceeding proof show that for a small category V, the application A t+ hA, f t+ hf is a contravariant functor h- : V --f Funct(V, S). We call h- the contravariant representation functor. Correspondingly, h- : V ---t Funct(Vo, S) is the covariant representationfunctor. Both functors have the property that the induced maps on the morphism sets are bijective. A full functor is a functor which induces surjective maps on the morphism sets. A faithful functor is a functor which induces injective maps on the morphism sets. A faithful functor is sometimes called an embedding. Thus the representation functors are full and faithful. Already in Section 1.8 we realized that the image of a functor is not necessarily a category. This, however, is the case if the functor S :V +9 is full and faithful. Obviously we only have to check whether for f : A --f B and g : C + D in V with S B = S C the morphism F g F f appears in the image of S.Since Mor9(SB, S C ) Mor,(B, C) and Mor,(SC, S B )E Mor,(C, B), there are h :B + C and k : C + B with S h = lgFBand S k = 19,. Since S ( h k ) = 19, = 91, and S ( k h ) = 19, = 9 1 , , we get hk = l c and kh = 1,. Thus F g S f = S ( g ) 1 9 B F ( f ) = S ( g ) S ( h ) 9 ( f ) = $(ghf)* The full and faithful functors are most important, as we want to show with the following example. Let S : V + 9 be full and faithful. Let

1.15

45

THE YO:NEDA LEMMA

be a diagram in %' which is carried over by 3 into the diagram

SC,

except for the morphism h. Assume that there is a morphism h in 9 making the diagram commutative. T h e question is, if there is also a morphism h' : C, -+ C, making the diagram in V commutative. F being full and faithful, we may take the counterimage h' of h for this morphism. Thus we decided the question for the existence of morphisms in 53' with particular properties in the category 9.

LEMMA 2. Let F : 53' -+ 9 be a fuk' and faithful functor. Let A? and a be diagram schemes and 9 : d -+ V and 9' : 9 -+ 23 be diagrams. Let & : d -+ LB be a functor which is byectiue on the objects such that the diagram 8

& -4.

g

.1

$1 .%

g 4 - 9

is commutative. Then there is exactly one diagram 8 : = 9.

-+V

such that

.FA? = 9'and 2 8

Proof. We define 8 on the objects of LB by 9,since & is bijective on the objects. For the morphisms of LB we define &? by the maps induced by 9' and F - I . Here we use that 3 is full and faithful. With this definition of the map 8 one verifies easily that 3? is a functor and that 8 satisfies the required commutativities.

Let V be a small category. Let 1 Y be a small full subcategory of S containing the images of all representable functors from V to S.In this case we can also talk about the representation functor h:V--+Funct(V,d). Correspondingly, we define a representation functor H from Funct(V,A) which is again a small category, into Funct(Funct(V, A),S). Both functors are full and faithful. T h e composition of H and h gives a functor, which is isomorphic to the evaluation functor @ : % .-+ Funct(Funct(%, A),S)

which is defined according to the evaluation functor @ : % x Funct(%, A)-+

S

1.

46

PRELIMINARY NOTIONS

This is implied by Corollary 1. Thus the evaluation functor Q,

:9 ? -+ Funct(Funct(W, A),S)

is full and faithful. Now we want to generalize the assertions of the Yoneda Lemma to functors. We consider functors 9, 9 : W -+ 9. With Mor9(P-, -) we denote the composed bifunctor from W x 9 into S with Mor9(9-, -)(C, D) = hlorg(SC, D) and

MordS-,

-)(J

g) = M o r d W , g)

For a natural transformation F : 9--t 9,let

denote the natural transformation which is defined by Mor9(pC, D)(f) = E MorB(9C, D). With these notations we obtain the following lemma.

f v ( C ) , where f

LEMMA 3.

The application

Mor,(F, 9)3

ct

MorB(p-, -)

E

Mor,(Mor&-,

-), Mora(S-, -))

is bijective. I t induces a bijection between the natural isomorphisms from 9 to 9 and the natural isomorphisms from MorB(9-, -) to MorB(9-, -). Proof. A natural transformation t,h : Morg(9-, -)+ MorB(9-, -) is Mora(9C, -) a family of natural transformations t,h(C): Mora( 9C, -) which is natural in C for all D E 9 (Section 1.14, Corollary). T h e natural transformations #(C) may be represented as Mor9(g,C, -) with morphisms FC : 9 C + 9 C by the Yoneda lemma. Thus it suffices to prove that FC is natural in C, if Mor,(qL’, D ) is natural in C for all D E 9. One direction may be seen if one replaces D by 9 C in the diagram --f

and if one computes the image of lBC. T h e converse is trivial. T h e assertion on the natural isomorphisms follows from the considerations

1.15

THE YOblEDA LEMMA

47

in Section 1.5-the isomorphism has to be tested only argumentwiseand from Corollary 2(b). We define an equivalence relation a n the class of objects in the following way. Two objects are called equivalent if the representable functors, represented by these objects, are isomorphic. By the Yoneda lemma this is the same equivalence relation as the one defined by isomorphisms of objects. Since in categories one considers only the exterior properties of objects, which are, of course, carried over to isomorphic objects, it makes sense to generalize the notion of a representable functor. A functor 9 : $? -+ S is called representable, if there is a C E $? and a natural isomorphism 9 E hC. Here the representing object C is only defined up to an isomorphism. This generalized notion leads to the following lemma.

LEMMA 4. Let 9 : V x 9 -+ S be a bifunctor such that for all C E $? the functor 9 ( C , -) : 9+ S is representable. Then there is a contravariant functor B : $? -+ 9, such that F E M[or,(B-, -). Proof. Let 9’ be a skeleton of 9. T o each C E $? there exists exactly one D E 53‘ with F ( C , --) E Mor,(D, -). Let us denote D by B(C). The natural isomorphisms F ( C , --) Mor,(D, -) are in one-one correspondence with the elements of a subset F‘(C, D)of S ( C , 0)by the Yoneda lemma. For each C E $?,this subset S ’ ( C , 0)is uniquely determined. By the axiom of choice, .we may assume that to each C E $? there is exactly one element c E F ’ ( C , D).(With the formulation of the axiom of choice we use, one has to form a disjoint union of the sets F ( C , D)with the equivalence relation C-C‘ o V C with c, c’ EF’(C, D).) Thus, for each C E $? there is a natural isomorphism hc : Mor,(D, -) --t 9 ( C , -). Let f : C-+ C’ be a morphisim in %‘.Thenby the Yoneda lemma there is exactly one morphism Bf: B(C’) + B(C) in 9 making the diagram hC

Morg(9(C), -) -hlor,(S,,-)

1

Mor9(9(C‘), -)

S(C, -)

-hC‘

IS(f,-)

S(C,-)

commutative. This uniqueness and the property of a functor of F imply that Bfg = 9 g B f and 9 1 c = lg(c).Thus 9 is a contravariant functor from %‘ to 9 with the required properties.

48

1.

PRELIMINARY NOTIONS

1.16 Categories as Classes In Section 1.2 we mentioned that a category may be considered as a special class. Now we want to specify this. First, we deal with the definition of a category that describes only the properties of the morphisms, but does not define the objects. This definition will be slightly narrower than the one given before. First we want to give the definition; then we want to investigate the connection with the definition given in Section I. 1.

A category is a class A! together with a subclass Y- C &? x A? and a map V 3 (a, b) Hab EA?

such that (1) For (i) (ii) (iii) (iv)

(2)

all a, b, c E &? the following are equivalent (a, b), (4c ) E Y(a, b), (ab, c ) E Y (a, bc), (b, c) E Y(a, b), (b, c), (a, bc), (ab, c )

E

7G/‘ and (ab)c = a(bc)

For each a E A! there are e, , e, E A! such that ( e l , a), (a, e,) and e,b = b, b’el = b’, e,c = c, c’e, = c‘ for all (el 3 b), (b’, e l ) , (e, 9 c), (c’, e,) Then e, and e, are called units.

E

Y-

E

(3) Let e, e’ be units. Then {a I (e, a), (a, e’) E

V)

is a set. It is easy to verify that the morphisms of a category (in the sense of Section 1.1) satisfy this definition. Conversely, one can get the objects of a category out of the class of morphisms if one assigns to each identity an element, called an object. This, however, does not determine the class of objects uniquely. In this sense the definition given here is narrower. Now we have to prove that each class satisfying the present definition occurs as a class of morphisms in a category (in the old sense). Let A!, Y- satisfy the given definition. We form a cntegory V (in the old sense) with the units e E A? as objects. Furthermore, we define Morop(e’, e) := { a I (e, a), (a, el) E V }

49

PROBLEMS

These morphism sets are disjoint. I n fact, if aE

Morrg(e’, e) n Mory(e**, e*)

then (e, a), (e*, a), (e, e*a), (e, e*) E 9‘-thus e = ee* = e*. Similarly, we get e‘ = e**. For a E Moru(e’, e), b E MorV(e**, e*) we have ( a , b) E V if and only if (ae’, e*b), (a, e’), (e*, b), (e’, e*) E V if and only if (e’, e*) E V if and only if e’ = e*. I n this case we have (e, ab), (ab, e**) E V ,thus a b More(e**, ~ e). Now it is easy to verify the associativity and the properties of the identities. T o get the connection with set theory as discussed in the appendix, we now define the category as a special class. A class 9is called a category if it satisfies the following axioms: (a) (b) (c) (d)

~ C U X U X U D ( 9 ) C ! l B ( 9 ) x !lB(9) B is a map For A’ = !lB(9),V = D ( 9 ) and 9 : V (l), (2), and (3) given above are satisfied.

+ A‘

the axioms

Obviously this definition is equivalent to the definition of a category given above. Problems 1.1.

Covariant representable functors from S to S preserve surjective maps.

1.2. Check whether monomorphisms [epimorphisms] in Ab and Top are injective [surjective] maps.

B is a dense map. (Hint: Use as a test object 1.3. In Hd each epimorphismsf : A the cofiberproduct of B with itself over A (see Section 2.6).) -+

1.4. Show: If Y : %‘ -+ 9 i s an equivalence of categories and j~ Q is a monomorphism, then Yf is a monomorphism.

B be an epimorphism and a right zero morphism. How many 1.5. Let f : A elements are there in Morrg(B, C)? Compute MorRi(P, P). -+

1.6.

Let A be a subset of a topological space (B, OB).

{XIX=AnY;

YEOB)

defines a topology on A , the induced topology. A C B, provided with the induced topology, is called a topological subspace of (B, OB).T h e topological subspaces of a topological spaces are (up to equivalence of monomorp hisms) exactly the difference subobjects in Top. Dualize this assertion. To this end, define for a surjective mapf : B -P C a quotient topology on C by I C .f-’(Z) E OD}

{zz c;

1.

50

PRELIMINARY NOTIONS

1.7. A subgroup H of a group G is a subset of G which forms a group with the multiplication of G. A subgroup H of G is called a normal subgroup if gHg-' = H for all g E G. Show that the subgroups [normal subgroups] of G are, up to equivalence of monomorphisms, exactly the difference subobjects [normal subobjects] of G in Gr. 1.8. I f f is an isomorphism, then f is a retraction. T h e composition of two retractions is a retraction. If f g is a retraction, thenf is a retraction.

1.9. If W is a category with zero morphisms, then the kernel of a monomorphism in Q is a zero morphism. 1.10. Let Q be a category with a zero object 0. Let A a product of A and 0. 1.11.

T h e diagonal is a monomorphism.

1.12.

If both sides are defined, then

1.13.

Let B : S

EQ,

then (A, 1"

,O(ASO))

is

!(A) C g - ' ( ( g f ) ( A ) ) +S

be defined by

B(A) = { X I X C A )

and

B ( f ) ( X )= f - l ( X )

then B is a representable, contravariant functor, the contruwnrinnt power set functor. 1.14.

Let 2 : S

+

S be defined by

1(A) = {X 1 X C A }

and

9(f ) ( X ) = f ( X )

then 2 is a covariant functor, the cowariunt power set functor. Is 2 representable ? 1.15. If 9 : S -+ S is a contravariant functor and f : 9(0)) { A is an arbitrary map, then there is exactly one natural transformation q : 9 -+ Mars(-, A) with v({ a}) = f. (Observe that Mors(B, 9({ a)))= (9({ --f

1.16. Let 9 : S --* S be a faithful contravariant functor; then there is an element b in F(2),which is mapped into two different elements of 9 ( 1 ) by the two maps 9 ( 2 ) -+ P(1).Here let 1 be a set with one element and 2 be a set with two elements. 1.17. (Pultr) Let 9 : S -+ S be a faithful contravariant functor, then there is a retraction p : 9 B, where 9' is the contravariant powerset functor. (By the Yoneda lemma, it is sufficient to prove that there exists a b E 9 ( 2 ) for which p(2)(b) is the identity on 2. Use problems 13, 15, and 16.) --f

1.18. In the category of Section 1.1, Example 14, the greatest common divisor of two numbers is the product, and the least common multiple of two numbers is the coproduct.

2 Adjoint Functors and Limits One of the most important notion:; in the entire theory of categories and functors is the notion of the adjoint functor. Therefore, we shall consider it from different points of view: as a universal problem, as a monad, and as a reflexive or coreflexive subcategory. T h e limits and colimits and many of their properties will be derived from the theorems which we shall prove for adjoint functors. This procedure was introduced by D. N. Kan. T h e paragraph on monads should be considered preparation for the third chapter. I n this field there is still fast development. With the means given here, the interested reader will be able to follow future publications easily.

2.1

Adjoint Functors

I n Section 1.15, Lemma 3 we de.alt with the question of what the isomorphism Mor,($---, -) Mor,(8-, -) means for two functors F and 3 ' .Now we want to investigate under which circumstances there -) Mor,(-, 9-). First, is a natural isomorphism Mor,(.F-, 9 : % 3 9 and 9 : 9 -+ % must be functors. Two such functors are called a pair of adjoint functors; 9 is called left adjoint to 9 and 9 is called right adjoint to .F if there is a natural isomorphism of the -) r Mar,(-, 8-) from q0x 53 into S. bifunctors Mor,(9-,

PROPOSITION I . Let the functor 9 : V + 9 be left adjoint to the functor 99 : 9 -+ %. Then 9is determined by 8 uniquely up to a natural isomorphism. Proof. Let 9 and F' be left adjoint to 9,then there is a natural Mor,%(S'-, -). Thus, by Section 1.15, isomorphism Mor9(9--, -) Lemma 3 we have 9 1; 9'. If there is a left adjoint functor to 9 which is uniquely determined up to an isomorphism, it will also be denoted by *9.If we pass over 51

2.

52

ADJOINT FUNCTORS AND LIMITS

to the dual categories V oand B0,then we get, from the considerations of Section 1.4, functors O p F O p = 9 0 :VO + 9 0 and Op9Op = 9 O : 9 O -+ Vo,and we have MorWpo(9O-,-) Margo(-, g o - ) Thus . gois left adjoint to Foand is uniquely determined up to an isomorphism by 9 O . Since 9 = goo,9 also is uniquely determined by 9 up to an isomorphism. Thus the properties of left adjoint functors are transferred to right adjoint functors by dualization. If there is a right adjoint functor to F which is uniquely determined up to an isomorphism, then it will also be denoted by F*.

COROLLARY 1 . Let the functors gi: V + 9be left adjoint to the functors 9$: 9 + V for i = 1,2. Let v : gl + g2be a natural transformation. Then there is exactly one natural transformation *v : F 2+ F1,such that the diagram MOT,(-, 8,-) gg Mor9(F1 -, -)

1

M o r g (*'P- ,-I

Morup(-,'P-)l

Mary(-, g2-) g Mor9(S2 -, -)

is commutative. If tp = idyl, then *tp = idFl natural transformations, we have *(a$) = *$*tp.

. For

the composition of

Proof. The first assertion is implied by Section 1.15, Lemma 3. The other assertions follow trivially. COROLLARY 2. Let V and 9be small categories. The category Funct,(V,9) of functors from V into 9 which have right adjoint functors is dual to the category Funct,(9, V ) of the functors from 9 into V , which have left adjoint functors.

PROPOSITION 2. A functor 9 : 59 + V has left adjoint functor ij and only if all functors Mor,(C, 9-)are representablefor all C E V . Proof. This is implied by Section 1.15 Lemma 4. Now we have to deal in more detail with the natural isomorphisms -) + Mor,(-, 9-) used in the definition of the adjoint functors. First we assume that q~ is an arbitrary natural transformation. Let objects C E V and D E 9 be given. Then : Mor9(F-,

v(C, 0) : Mor9(9C, 0)+ Morq(C, SO).

If we choose in particular D

=

S C , then we get a morphism

T(C, %C)( I s = ) : c + 599-c

2.1

53

ADJOIN” FUNCTORS

for all C E%?. These morphisms form a natural transformation @ : Idw + 99.In fact, iff : C --+ C’ is a morphism in V, then the diagram Mor(SC, S C )

Mor(.F/,.FC‘)

Mor(SC,F/) b

Mor(%C, %C)t

Mor(%C’, F C )

is commutative. Thus @(C’)f = Mor(f, 9 S C ‘ ) v(@, FC’)(lgc,) = a(C, S C )Mor(Sf, % C ) ( l . ~ ~ e ) = y(C, S C ’ ) ( S f )= p(C, %C’)

=

Mor(.FC, Sf)(lFc)

Mor(C, 3 S f ) p(C, F C ) ( 1 g c ) = %sf@( C)

Conversely, if @ : Id, define a map

+ 99

is a natural transformation, then we

v : MorQ(FC,D)3 f w !qf@(C)E Morw(C, 9D) I t is natural in C and D because it is a composite of the maps B : Morg(FC, D) -+ Morw(9SC, 9D)

and Mor(@C, BD) : Mory(9SC, 9D)-+Mory,(C,9D)

But both maps are natural in C and D.

LEMMA.Let F : % + 9 and 3 : 9 -+ V be functors. The application Mor,(Id, , 9%)3 @ t+ ’3’4- E Mor,(Mora(S-,

-), Morw(-, 9-))

is bijective. The inverse of this application is Morf(Morg(S-,-),

More(--, 3 - ) )3 p ) ~ ~ ( -9 ,- ) ( 1 ~ - ) E Morf(Idg, 9%)

Proof. Let @ be given, then 9(lFC) @(C) = 9 9 ( l c ) @(C) = @(C). Let be given, then %f(V(C,.FC)(IFC))= Morw(C, ?f)V(C, S“c>(L3w) = V(C, D)MOrad(F‘C,f)(lFc) = V(C, D ) ( f )

Dual to the lemma one proves that Mor,(99, Ida) g Morf(MorSl(-, 9-),Morg(S-, -))

54

2.

ADJOINT FUNCTORS AND LIMITS

With the same notations as before, we have the following theorem.

COROLLARY 3. The functor 9 : V ---t 9 if left adjoint to g : 9 + V if and only if there are natural transformations @ : Idy + 3 9 and Y : 9 3 -+ Id, with ($Y)(@$) = id, and ( Y 9 ) ( 9 @= ) id,. COROLLARY 4. Let 9 be left adjoint to ’3, then the maps

3’ : Mor,(.FC, D)+ Moryp(%SC, BD)

are injective for all C E V and D E 9.

2.1

ADJOINT FUNCTORS

55

Proof. By the considerations preceeding the lemma, the isomorphism More(-, 9-) is composed of the morphisms Mor9(9-, -) 9 : Morp(S-, -)

---f

Morv(9S-, 9-)

--f

Moryp(-, 9-).

and

Mor@(9S-, 9-)

COROLLARY 5. Let the categories V and 9 be equivalent by 9 : V -+9 99 and Y : 9 9 I d a , then 9 is left and 9 : 9 -+ V, @ : Idv adjoint and right adjoint to 9.

Proof. @9 and 9" are isomorphisims. Consequently, (9!P)(@B)and ( Y 9 ) ( 9 @are ) also isomorphisms. Thus, zh,p and qx,b are isomorphisms and also IJJ and 4.

PROPOSITION 3 . A functor 9 : 2? + 9 is an equivalence if and only if9 D. is full and faithful and if to each D E 9 there is a C E V such that 9 C Proof. T h e conditions are easy to verify if 9 is an equivalence. Now let 9 be full and faithful and let there be a C E V to each D E 9 such that F C 9.We consider the functors % : V' + V and 9 : 9 + 9' which are equivalences between V and 9 and the corresponding skeletons V' and 9 respectively. Obviously, 9 is an equivalence if and is an equivalence. 99s is full and faithful only if 9 9 X : V' + 9' since any and all objects of 9'appear already in the image of 99s) two isomorphic objects in 9' are already equal. T h e considerations on the image of a full and faithful functor in Section 1.15 show that different objects of %' are mapped to different objects by 99s. Thus 39sis bijective on the class of objects and on the morphism. T h u s the inverse map is a functor and 99% is an isomorphism between V' and 3'. In Corollary 3 we developed a first criterion for adjoint functors. Before we develop further criteria and investigate in more detail the properties of adjoint functors, we want to give some examples of adjoint functors.

Examples

1 . Let A E S. Forming the product with A defines a functor A x S + S.There is a natural isomorphism (natural in B, C E S) Mors(A x B, C )

- :

Mors(B, Mors(A, C ) )

2. Let M o be the category of monoids, of sets H with a multiplication H x H + H , such that (h,h,) h, = h,(h,h,) and such that there is a

56

2.

ADJOINT FUNCTORS A N D LIMITS

neutral element e E H with eh = h = he for all h E H , together with those mapsf withf(h,h,) = f ( h , ) f ( h , ) and f ( e ) = e. Given a monoid H , we define a unitary, associative ring by Z ( H ) = {f I f~ Mors(H, Z) and f ( h ) = 0 for all but a finite number of h E H } We define (f + f ’ ) ( h ) = f ( h ) + f ‘ ( h ) . Then Z(H) becomes an abelian group. T h e product is defined by ( f f ’ ) ( h ) = C f ( h ’ ) f ’ ( h ” ) where the sum is to be taken over those pairs h’, h“ E H with h‘h” = h. Since H is a monoid, we get a unitary, associative ring Z(H). Furthermore, Z( -) : Mo -+ Ri is a covariant functor. Now let R E Ri and let R’be the monoid defined by the multiplication on R, then also -’ : Ri -+ Mo is a covariant functor. There is a natural isomorphism

that is, the functors constructed above are adjoint to each other. This and other functors will be investigated in more detail in Chapter 3.

3. The following is one of the best known examples which, in fact, led to the development of the theory of adjoint functors. Let R and S be unitary, associative rings. Let A be an R-S-bimodule, that is, and R-leftmodule and an S-right-module such that ( a s ) = (ra)sfor all I E R,s E S, and a E A. T h e set Mor,(A, C) with an R-module C is an S-left-module by ( $ ) ( a ) = f ( a s ) . Mor,(A, -) : ,Mod .+ ,Mod is even a functor. T o this functor there is a left adjoint functor A 0,- : ,Mod -+,Mod called the tensor product. Thus there is an isomorphism

which is natural in B and C. Actually this isomorphism is also natural in A.

2.2 Universal Problems Let us consider again Section 2.1, Example 2. For each monoid H the natural tranformation Id,, -+ (Z( -))‘ induces a homomorphism of monoids p : H -P (Z(H))‘which assigns to each h E H the map with f ( h ’ ) = 1 for h = h’ andf(h’) = 0 for h # h’. Let us denote this map by fh . Now if g : H -P R is a map with g(h,h,) = g(h,) g(h,) and g(e) =

2.2

57

UNIVERSAL PROBLEMS

1 E R, then there is exactly one homomorphism of (unitary) rings g* : Z ( H ) -P R such that the diagram H

2E(H) R

is commutative. In this diagram we have morphisms of two different categories. In fact, p and g are in M o and g* is in Ri. Correspondingly, Z ( H ) and R are objects in M o and also objects in Ri. Furthermore, we composed a homomorphism of rings g* with a homomorphism of monoids p to a homomorphism of monoidsg. We want to give a structure in which these constructions are possible. Let V and 9 be categories. Let a family of sets {Mory(A, B ) 1 A E %‘, B E 9}

be given together with two families of maps E %‘,B E 9

Morv(A, A’) x Morv(A’, B ) + Mory(A, B),

A, A’

Morv(A, B’) x Mor&?’, B ) + Mory(A, B),

A E %, B’, B E 33

As usual we write these maps as compositions, that is, iff E Mory(A, A‘), v E Mory(A’, B), a‘ E Mor,(A, B’), and g E Morp(B’, B), then we denote the images of ( f , v) and (v’,g ) by vf and gv’ respectively.

LEMMA1. The disjoint union of the classes of objects of V and 9 together with the family {Morop(A, A’), Mory(A, B), MorB(B, B’) I A , A‘ E V, B, B‘ E 9)

of sets, which we consider as disjoint, and together with the compositions of V and of 9 and the above dejined compositions form a category V ( V ,9), the following hold for all A, A’, A” E V , B, B’, B” E 9 and for all f E Mory(A’, A), f ‘E Mory(A’’, A’), v E Morv(A, B), g E Mor9(N, B’), and g’ E Mor,( B’, B”) (1) (vf1.f‘ = v ( f f ‘ )

(2) (g’g)v = v ’ ( g 4 (3) ( P ) f= g(vf 1 (4) l s v = v = vl’4

2.

58

ADJOlNT FUNCTORS AND LIMITS

Proof. It is trivial to verify both axioms for categories if we set MorV-(,,a,(B,

4=

@.

If Lemma 1 holds, then we call the category Y(%, 9)directly connected category. T h e family of sets Mory(A, B) is called a connection from % to 9.

If we want to express our example with this structure, then we first have to define a connection from Mo to Ri. For H EMOand R E R ~ , we define Morv(H, R ) = Mor,,(H, R’), where R’ is the multiplicative monoid of R. By using indices we can make Mory(H, R ) disjoint to all morphism sets of Mo. T h e compositions are defined by the composition of the underlying set maps. T h u s we get a directly connected category Y(Mo, Ri). Now to each H E Mo there is a morphism p : H -+ Z ( H ) such that to each morphism g : H R for R E Ri, there is exactly one morphism g* : Z ( H ) + R making the diagram ---f

H

Z(H)

R commutative. In the general case, a directly connected category gives rise to the following universal problem. Let A E %. Is there an object U ( A )E 9 and a morphism pA : A + U(A), such that to each morphismg : A -+ B for B E 9 there is exactly one morphismg* : U(A) ---f B making the diagram A %U ( A )

B

commutative ? A pair (U(A), P A ) satisfying the above condition is called a universal solution of the universal problem.

LEMMA 2. Let V ( V ,9) be a directly connected category. The universal problem defined by A E V has a universal solution ;f and only if the functor Mory(A, -) : 9 + S is representable.

Proof. If (U(A), pA) is a universal solution, then by definition Mor(p, , B) : Mor,( U ( A ) ,B) Mor,(A, B). Furthermore, by the Yoneda lemma, Mor(pA , -1 : Mor~-(w,m(U(A), -1

-

Morvcw.a)(A, -1

2.2

59

UNIVERSAL PROBLEMS

is a natural transformation. Conversely, if @ : Morg( U(A),-) :z Mor,(A,

-),

then again by the Yoneda lemma @ = Mor(@(U(A))(l.(,)), -) U(A) E 9. But this means that the natural transformation Mor9( U(A),-)

since

Mory(A, -)

maps the morphisms of Mora( U(A), .B)into MorV(A, B) by composition with @( U(A))( 1U ( A ) ) . Thus ( U ( A ) ,@( U(A))(lu(A))) is a universal solution of the problem. This lemma implies immediately that a universal solution of a universal problem is uniquely determined up to an isomorphism. A directly connected category V(%, 9)is called universally directly connected if the corresponding universal problem has a universal solution for all A E V. Often the connection for a directly connected category is given by a functor as MorV(A, B) := Mory(A, 923) Then we also write Vcyp(V, 9).Because of the functor property of B each covariant functor B defines a connection. Similarly, each functor 9: V --t 9 defines a connection by Morv(A, B) := N[orB(FA, B )

LEMMA 3. The directly connected category V ( U , 9) is universally directly connected ;f and only if there isfunctor 9 : V + 9 such that there exists a natural isomorphism MOTV(-, -) E Morg(2F-, -). Proof. The lemma follows immediately from Lemma 2 and Section 1.15, Lemma 4.

THEOREM 1 . Let 9 : 9 -+V be a cooariant functor. The following are equivalent: 9 has a left adjoint functor 2F : V -+ 9. ( 2 ) The directly connected category Vy,(%, 9 ) is universally directly connected. (1)

In this special case we want to reformulate the universal problem using the definition of the connection. Let 9 : 9 -+V be a functor. Let A E V. We want to find an object %A E 9 and a morphism pA : A -+ 9 9 A

60

2.

ADJOINT FUNCTORS AND LIMITS

such that to each morphism g : A --t 3 B for each B E 9 there is exactly one morphism g* : F A -,B which makes the diagram

commutative. Here it ecomes clear t,,at not g* is composed with p A but 3g*. The example with which we started at the beginning of this section has exactly this form. Let two categories V and 9 be given. Let a connection {Mory(B, A ) I B E 9, A E g}

be given such that V ( 9 ,V) is a directly connected category. We also denote this category by -Y-'(%', 9)and call it universely connected category. Observe that now Mory,(Y,P)(B,A) is not empty in general, but that Morv*(,,,)(4 B) = 0 . Let V ( V ,9) be an inversely connected category. Here again we define a universalproblem. Let A E V. Is there an object U(A)E 9 and a morphism p A : U ( A )-+A such that for each morphism g : B + A for all B E 9 there is exactly one morphism g* : B U(A) making the diagram B --f

8.1

\

U ( 47 A

commutative ? A pair ( U ( A ) ,p A ) satisfying the above condition is called a universal solution of the universal problem. If the universal problem in V ( V ,9)has a universal solution for all A E V, then Y ' ( V ,9)is called universally inversely connected. Thus we get a new characterization for pairs of adjoint functors F : V --t 9 and 9 : 9 + V.

THEOREM 2. Let categories V and 9 and a connection be given such that Y ( V ,9)is directly connected and V(9,V) is inversely connected with the given connection. Then the following are equivalent: (1) Y ( g ,$9) is universally directly connected and V(9,U) is universally inversely connected.

2.3

61

MONADS

(2) The morphism sets of the connection are induced by a pair of adjoint functors

F and 3 as

MOTy(-,

-)

g

MorB(9-,

-)

Morup(-, B-)

Proof. This assertion is implied by Lemma 3 and the dual of Theorem 1.

2.3 Monads

a,

Let d , %‘) and 9 be categories, 9, 9‘: d -28, 3) 9’) 9”: 9 ? + V , a n d X , X ’ : V + - t b e f u n c t o r s , a n d p , :9+9’,#: 3+3’, $’ : 9’--t 3“, and p : Z‘ --t 3Ea’ be natural transformations. In Section 2.1 we saw that also $9: 3 9 4 9’9and Z#: 3Ea3 --.+ Z3’with ($F)(A) = t,h(F(A)) and (Xt,h)(B)= X ( $ ( B ) )are natural transformations. With this definition one easily verifies the following equations:

where the last equation follows from the fact that # is a natural transformation. Now let S : V ---f 9 and 9 : 9 + V be a pair of adjoint functors with the natural transformations @ : Idy --t 99 and Y : 9 3 + Id9 satisfying the conditions of Section 2.1, Theorem I , We abbreviate the functor 9 9 by X = 99.Then we have natural tranformations r=@:ldy-+Z

and

p=B?PF:22-+2

With these notations we obtain the following lemma.

LEMMAI.

The following diagrams are commutative:

2.

62

ADJOINT FUNCTORS AND LIMITS

Proof. We use Section 2.1, Theorem 1 and obtain from the definitions p(&)

= (9Y9)(@99) = ((BY)d)((@9) 9) = ((BY)(@9))B=

idg%

=

id#

p ( x € ) = (9Y9)( 99@)= (B(Y9))(9(9@))

p(@)

=

B((YS)(9@))= 9 i d s

=

( S Y 9 ) ( 9 $ S 9 9= ) 9(Y(Y99)) 9 = Y(Y(99Y))d

= id&

= (9Y9)(999YB) = p(Hp)

A functor 2' : V + V whose domain and range categories coincide is called an endofunctor. An endofunctor i%? together with natural transformations : Id, + 2 and p : 2'# --f 2 is called a monad if Lemma 1 holds for (2, e, p). Other terms are triple or dual standard construction. T h e dual terms are comonad or cotriple or standard construction. T o explain the name, one notes that a monoid is a set H together with two maps e : {a}+ H and m : H x H + H such that the diagrams ex h

mx h

H - H x H hxe/

H

X

\ H

H x H x H - H x H hxm/

/m

~

H

1.

HXHA

are commutative, where we identified {a} x H with H. Observe, however, that in the definition of the product we did not use the product of the endofunctors but their composition. T h e term monad was proposed by S. Eilenberg because of this similarity. Now we want to deal with the problem of whether all monads are induced by pairs of adjoint functors in the way we proved in Lemma 1. We shall see that this is the case, but that the inducing pairs of adjoint functors are not uniquely determined by the monads. There are, however, two essentially different pairs of adjoint functors satisfying this condition and having certain additional universal properties. These pairs were found by Eilenberg, Moore, and Kleisli. We shall use both constructions with only minimal changes.

THEOREM 1. Let (2, Q, p) be a monad over the category V. There exist pairs of adjoint functors 9 ,:V -+V, ,5, : Q, +V and 949" :V +VH, Y H:V m + V inducing the given monad. If 9 : V + 9, g : 9 --+ Q

2.3

63

MONADS

is another pair of adjoint functors inducing the given monad, then there are uniquely determined functors X and 9 making the diagram

commutative. Proof. First we give the construction of 9 *, F ,, and V , . The objects of V, are the same as the objects of V. Let A, B E V. T h e morphisms from A to B in V, are the morphisms f : &A -+ &B for which the diagram XXA CAI

XA

3X X B

- 1.. f

XB

is commutative. By using indices we can determine that the morphism sets in V, are disjoint. The compositions are defined as in V . Then Vx is a category because & is a functor. , and F, by 9,A = A, 9* ' f = &f and We define the functors 9 , is a functor. The functor F'A = X A , Fx f = f . Trivially, F properties of 9 ,are implied by the fact that p is a natural transformation. Furthermore, we have X = F,9,. To show that 9 ,is left adjoint to F , we use Section 2.1, Corollary 3. by YA = p A : Let CP = E : Idy -P &. Define Y :F Y ,, -+ Id,, X X A -+%A considered as a morphism from &A to A in V,. !PA is a morphism in V, because of p ( X p ) = p(p&). Y is a natural transformation because of the hypotheses on the morphisms in V, . Then we have for objects A E V and A E % ,' respectively, (Y%P)(%V@)(4 = ( Y % f W ) ( % f 4 A )= )r(4X44=

l,A

=

19,A

44 €#(A) = 1,A

=

l,,"

and

(.?P)(@~.)(4 = (~,Y(A))(@Y,(A))

=

Since p = F,Y9*, the monad ( X ,E , p) is induced by the pair of adjoint functors 9 ,and 9-,. T , is faithful by Section 2.1, Corollary 4, since all objects of V, are in the image of 9 ,.This also follows directly from the definition.

2.

64

ADJOINT FUNCTORS AND LIMITS

Now we give V", 9'",and F". The objects of V" are pairs (A, a) where A is an object in V and a : H A -+ A is a morphism in V such that the diagrams

are commutative. The morphisms from (A, a) to (B, 8) are morphisms f : A --t B in V with the diagram XA

A-B

"f

XB

f

commutative. The compositions are defined as in %'. Then V" is a category. The functors 9 '" and Y* are defined by Y"A = ( S A ,P A ) , 9 ' " " ' = Hfand F"(A, a) = A , F"f = f. Trivially, Y" is a functor. ( H A , p A ) is an object of V" because (#, E , p ) is a monad. Sfis a morphism in V" because p is a natural transformation. Furthermore,

S

=

Y"9".

We use again Section 2. I , Corollary 3 to show that 9 '" is left adjoint to F". Let @ = E : Id, --t S.For each object (A, a) in V", we define a morphism Y ( A ,a) : Y"Y"(A, a) -+ (A, a) by a : H A --t A. Y(A, a) is a morphism in V" because of the second condition for objects in W" and because Y*Y"(A, a) = (#A, PA).Y is a natural transformation. I n fact, we get a commutative diagram XXB

NB

V

XB

"P

+

XB

2.3

65

MONADS

where f is a morphism from (A, a) to (B, 8). For objects A E V and (A, 01) in 5YM we get (Y9Jq(9*@)(A)= (Y9fl(A))(Y*@(A)) = p ( A )X € ( A = ) 1,

= lyap,

and ( Y J V ) ( @ F * ) ( Aa), = (Y'"Y(A, a))(@F*(A, a)) = a@)

= 1, = l F q A , u )

Then we have Y H Y Y J 4 ( A =) Y"Y(&A, p A ) = Y s ( p A ) = p ( A ) , thus the monad (3, E , p) is induced by the pair of adjoint functors 9 " and .Y#. By definition YJ1" is faithful. Now let F : V + 9 be left adjoint to 92 : 9 + %? with the natural transformations @' : Idv + 9 2 9 and Y : F g + Id, constructed in 2 9 ,E = @', and p = 92Y9, Section 2.1, Theorem 1. Let A? = 9 that is, let the monad (Z,E , p ) be induced by the pair 9 and 92. We define the functor X : '&# + 9 by X A = S A , Let f : &A -+ &B be a morphism of objects A and B in W H . Then we set

Xf= ( Y F B ) ( S j ) ( F @ ' A ) . By the definition off we havef(pB) = ( p A ) ( Z f ) .Using the definition of get (9')(992Y'FA) = (F9Y'SB)(S92Ff),thus Ff =

p, we

( . ~ f ) ( ~ 9 2 Y ' S A ) ( 9 9 2 ~= @ ' (A~) 9 2 Y ' F B ) ( S g F f ) ( ~ 9 2 9=~ ' A ) FgXf. Since Y' is a natural transformation, we get (!P'SB)(Sf) = ( Y ' F B ) ( F G X f )= (Xf)(Y'FA). Now let g : S B + S C be another morphism in V , (Jfg)(.xf)

=

. Then

(Zg)(YFw?f)(*@'4 = ('ulFC)(Fg>(%f)(F@'4 =Xgf

Thus we get that X is a functor. We have X Y & ( A ) = S ( A ) for A E V and X9&(f) = X(Pf) = (U'FB)(SSFf)(F@'A) = (YFB)(F@B)(Ff) = Ff

for f E V. Thus we get .XYX = 9. Furthermore, 92XA = 92SA = = F x A and

&A

SXf

=

( S Y F B ) ( P f ) ( 2 f @ ' A=) (pB)(.@f)(X€A)

= f(/LA)(P€A) = f = Yzf

hence '3% = F x . T o prove the uniqueness of X , we assume that there is another functor

2.

66 X' : V"

-,9

ADJOINT FUNCTORS AND LIMITS

which has the same factorization properties. Then

X ' A = F A = X A because 9'" is the identity on the objects. Let f : %'A .+ %'B be a morphism of objects A and B in V, .Then SXf = Fsf = SX'J I n particular 9 S X f = FSX'j. T h u s we get a commutative diagram

FBFA

SSXf

SBSB 1.m

Y..%Al 8

9A-SB

as well for g = Xf as for g = X y . Y ' F A being a retraction we get Xf = X'f thus X = T'. Now we want to construct the functor 9. Let D E 9 be given. T h e n we have a morphism S Y D : S S S D + SD. Now (SD,S Y D )is an object in V" because the diagrams SD

\

1 c9D/

VY'D

HBD

and

ZXBD

HVY'D

BD

ZBD [SY'D

PSD

SY'D

X3D

BD

are commutative, the first diagram because E = Of ,the second diagram 9 9and Y(Y'9$9) = Y(FSY).Thus we define because i%? = $ 9 D = ($9D,' 9 Y D ) . Let f :D .+ D' be a morphism in 9. Then the diagram BFBD SY'D1

BD

9.%9 f

B9BD'

-

1SY'D'

Sf

3D

is commutative. Consequently, $9f is a morphism in V". We define 9 f = Sf. Then 9'is a functor and we have 9 9 A = (%'A, P A ) and 9 . F f = i f f . Furthermore, we have .Fx9D

= .F'(3D,

Hence, 99 = 9 ' " and Y"S?

SYD) =

8.

and

.FJr"2'f

=

Bf

2.3

67

MONADS

We remark that because of XYA

= X p A = (Y'FA)(FSYFA)(F@'SFA) = (Y9A)(9(9P')(@'S)9A =) Y ' 9 A = Y ' X A

and Y9D

= Y(YD,S Y ' D ) =

SY'D

=Y

Y D

we have X Y = Y.X and Y 9 = 9 Y , where Y is the morphism from 9&T" to Id,,, and from YXY"to Id,# respectively. To prove the uniqueness of 3 let 2': 9 -+ g,, be another functor with the required factorization properties. T o prove that Y and 9' coincide on the objects, we first show that Y 9 ' = Y Y ' , which at any rate is true for 9. For this reason, we consider the two commutative diagrams

P'Y'D

Y'FSD

.I

*y'D

and

the objects and the vertical Because of 2'9'3= 9"B = YJL"Y"Y' morphisms in both diagrams are the same. Furthermore, 9-XYYJf =

=

SY'F

= 9-%9'Y,F

Since F Xis faithful, we also have

Y9'S

= YYH = P

Y'9

and

YYFSD

=Y

'Y9SD

that is, the upper horizontal morphisms in both diagrams coincide too. But since 9 Y ' D is a retraction, and retractions are preserved by functors, we also get 2 ' Y D = Y Y ' D , hence 9 ' Y = Y 9 ' . Let D E B and Y ' D = (A, a). Then A = F"(A, a) = 9 # Y ' D = BD and

68

2.

ADJOINT FUNCTORS AND LIMITS

hence 9 ' D = 9 D .Now let f : D + D' in 9 be given, then F"9f = 3 f = F"9'f. Since F" is faithful, we get 9 f = 97; thus, 9 = 9'. This proves the theorem. are called S algebras and the objects The objects of the category VJ1" of the form 9'"(A) are called free &' algebras.

COROLLARY. In the diagram of Theorem 1 the functors F", FJ1", and X are faithful. If one of the functors 2,9'",Y", or 9is faithful, then all these functors are faithful. Proof. The constructions of the proof of Theorem 1 imply that FH and F" are faithful. Because F" = 3Z,A? is also faithful. If X is faithful, then 9 is faithful, because X = 39.Now assume that 9 is faithful, then by Section 2.1, Corollary 4 the functor &' = 3 9 is faithful. Replacing 5 by the functors YJ1" and 9 ' " respectively, in both conclusions completes the proof. LEMMA 2. Let (S, c, p) be a monad over the category V, and let (A, a ) be an X algebra. Then there is a free 2 algebra (B,/I) and a retraction f : B -P A in V, which is a morphism of S algebras. Proof. By A

HALA a : X A + A is a retraction. Furthermore, S algebra. By

p : &'&'A

-P

&'A is a free

H H A 3H A

LAl

1.

HAAA

a is a morphism

of X algebras.

It is especially interesting to know under which circumstances the functor 2 : 9 + %" constructed in Theorem 1 is an isomorphism of categories. In this case one can consider 9 as the category of X algebras. A functor 9? : 9 --f 3 ' will be called monadic if 92 has a left adjoint

2.3

MONADS

69

functor F such that the functor 9 : 9 -+ VJI"defined by the monad B F = LPis an isomorphism of categories. Before we start to investigate this question in more detail, we need some further notions. First we want to make an assertion on the way functors behave relative to diagrams. Let B : 9 -+ V be a covariant functor. Let & be a categorical property of diagrams (e.g., f :A -+ B is a monomorphism, D is a commutative diagram, B -+ D is a product of the diagram 0). Assume that with each diagram D in V with property &, the diagram g(D)in 9 also has property @. I n this case one says that B preserves property &. Assume that each diagram D in $7 for which the diagram 9(D)in 9 has property & has itself property &, then we say that B rejects property &. Let D be a diagram in V with property & and with the additional property that there is an extension D" in 9 of the diagram g(D)with the property &*. If under these conditions, there is exactly one diagram extension D' of D in $7, with B(D') = D",and if this extension has property &*, then we say that B creates the property &*. A simple example for the last definition is the assertion that the functor B creates isomorphisms. This assertion means that to each object C E V and to each isomorphismf" : g ( C ) -+ C" in 9 there is exactly one morphism f' : C -+ C' in % ' with B(f') = f" and S(C')= C", and that then this morphismf' is even an isomorphism. T h e property & says only that the diagram D is a diagram with one single object and one morphism. T h e property &* says that the only morphism of the diagram with two objects, which is not the identity, is an isomorphism. T h e functor (5. of Section 2.4, Theorem 2 is an example of a functor which creates isomorphisms. In this simple case one even omits the specification of property @. A pair of morphisms fo ,fl: A -+ B is called contractible if there is . a morphism g : B -+ A such that f o g = 1, and flgjo = f1dl Let h : B -+ C be a difference cokernel of a contractible pair fo ,fi : A -+ B, then there is exactly one morphism k : C -+ B with hk = lc and kh = flg. For fig : B -+ B we have ( f l g ) f o = ( f l g ) fl . Since h is a difference cokernel of ( fo ,fl), there is exactly one K : C -+ B with kh = flg. Furthermore, we have hkh = hflg = hfog = hl, = l,h, and thus hk = 1, because h is an epimorphism. Conversely, let fo ,fl : A -+ B be a contractible pair with the morphism g : B -+ A. If h : B -+ C and k : C + B are morphisms with hf, = hfl , hk = l,, and kh = f,g, then h is a difference cokernel of (fo,fl). I n fact, if x : B -+ U is a morphism with xfo = xfl , then x = xfog = xflg = xkh. If x = yh, then xk = y. Thus, a difference cokernel of a contractible pair is a commutative diagram

70

2.

ADJOINT FUNCTORS AND LIMITS ‘6

B

tB

h

This implies the following lemma.

LEMMA 3. Each functor preserves diference cokernels of contractible pairs. Recalling the definition of an Z algebra for a monad (8, E , p), we see immediately that ( A , a) is an Z algebra if and only if the diagram

I

A

##A

‘A

tA

is commutative, that is, if a is a difference cokernel of the contractible pair (PA,*a). Let 9 : 9 -+ V be a functor. A pair of morphism f o ,fl : A --t B in 9 is said to be %-contractible if ( 9 f o , 9jl) is contractible in V. 9 creates digerenee cokernels of 9-contractible pairs if to each $-contractible pair fo,f l : A --+ B in 9 for which ($f0 , 9fl) has a difference cokernel h’ : 9B --f C’ in V, there is exactly one morphism h : B -+ C in 9 with Bh = h’, and if this morphism h is a difference cokernel of ( fo ,fl).

LEMMA 4. Let 9 : !2 -+ V be a monadic functor. Then 9 creates dz&rence cokernels of 9-contractible pairs.

2.3

71

MONADS

Proof. For a monad (X', E, p) we can assume 9 = %" and '3 = FJp. Letf, ,fl : (A, a) (B, 18) be a Y#-contractible pair, and letg : B A be the corresponding morphism. Assume that there is a difference cokernel h : B -+ C of fo ,fl : A -+ B (fi= 9-"fi).Then also Z h is a difference cokernel of ( X ' f , , Zfl).Thus, we get a commutative diagram --f

--f

A

A

fa

fa

:B

-

,B

h

h

*C

fi

where y : X'C + C is determined by the factorization property of the difference cokernel. Thus the first condition for an Z algebra holds for (C, Y). Since p : X'X' + X' is a natural transformation, pC : X Z C -+ X'C is uniquely determined by p A : X ' X A + Z A and p B : Z Z B + X'B as a morphism between the difference cokernels. T h e commutative diagrams

and

induce a commutative diagram

72

2.

ADJOINT FUNCTORS AND LIMITS

using f,, ,f i together with the usual conclusions for difference cokernels. Thus (C ) is an X algebra. Since 5 'y" is faithful, the morphism h in V" is uniquely determined by the morphism h in V. Furthermore, h is a morphism of X algebras with hf, = hf, . Now let k : (B, j3) + (D, 6) be another morphism of X algebras with kf, = kf,; then there exists exactly one morphism x : C + D in V with k = xh. Thus S k = S x S h . But since X h is a difference cokernel of Xf,and X f , , we get again, with the usual conclusions for difference cokernels, that 6 S x = xy. Thus, x is a morphism of Z algebras. This proves that h : (B, j3) -+ (C, y ) is a difference cokernel in V H ,

THEOREM 2 (Beck). A functor 9 : 9 -+ V is monadic i f and only i f 9 has a left adjoint functor 9, and i f 99 creates dijfuence cokunels of Y-contractible pairs. Proof. Because of Lemma 4, it is sufficient to prove that a functor 93, which has a left adjoint functor S, and which creates difference cokernels of 9-contractible pairs, is monadic. Here it suffices to construct an inverse functor for the functor 3' of Theorem 1. Let (A, a) be an S algebra with Z = 99.Then P A , S a : %&A -+ X A is a contractible pair with the difference cokernel a :X A +A. Since S ( Y 9 A )= p A and '??(.For) = #a, the pair Y ' S A , 9 a : 9 S A + 9 A is a 9-contractable pair which has a difference cokernel in V. The hypothesis implies that there is exactly one difference cokernel a : 9 A 4 C in 9 with 9, = a and SC = A. We define -%''(A, a) = C. Iff: (A, a) ---t (B, j3) is a morphism of X algebras, and if 9 ' ( B ,j3) = D and b :9 B -+ D is the difference cokernel of (??"9B, 9 j 3 ) then , the commutative diagram Y '.%A

S ~ S A7 FA 0 c

implies the existence and the uniqueness of the morphism g with 9 ( g ) = f . Let 9' f( ) = g. Since g is defined as a morphism between difference cokernels, 9' is a functor. Now we verify that 3'9' = Idv* and 9'9 = Id,. We have 3'3''(A, a) = (9C,S Y ' C ) = (A, BY'C). Since Y is a natural transformation, the diagram

2.4

REFLEXIVE SUBCATEGORIES

73

with (Y = S a and A = SC is commutative. S Y F A = p A and (SY'C) (SLY) = &A) = a ( 8 a ) and the fact that X u is an epimorphism as a difference cokernel imply a = 9 Y C . Furthermore, we have 99'f( ) = 9 ( g ) = S(g) = f , where g is chosen as above. Then 9 ' 2 ( C ) = 2 ' ( S C , SY'C). Since 9 Y C is a difference cokernel of the contractible pair 3 Y 9 9 C , 999Y'C : 99.%'3c --+ %S$??C

(the corresponding morphism is @ ' S S C )the , morphism Y'C : S S C --t C is a difference cokernel of (Y'.FSC, FSY'C) because of the hypothesis on 9.Thus, 9 ' 9 C = C. Furthermore, 9'9f = 9'Sf.Since the diagram

99c-cY'C 5.fl

99D

-If Y'D

D

is commutative, and since f is a morphism between difference cokernels, we have 2"gf = f.

LEMMA 5. Let S : 9 --+ V be a functor which creates dtfference cokernels of 3-contractible pairs. Then S creates isomorphisms. Proof. Let g : C + D be an isomorphism in V and let C = 9 A with A E 9. Then 1, , 1, : A --t A is a %-contractible pair with the difference cokernel g : C ---t D in V . Thus there is exactly one f : A --f B with Sf = g. Furthermore, f is a difference cokernel of I,, 1, : A --+ A. But also 1, : A -+ A is a difference cokernel of this pair, consequently f is an isomorphism in 9.

2.4

Reflexive Subcategories

Let 9 be a category and V a subcategory of 9. Let d : V -+9 be the embedding defined by the subcategory. V is called a reflexive subcategory, if there is a left adjoint functor 9 : 9 --t V to 8.T h e functor 9

74

2.

ADJOINT FUNCTORS AND LIMITS

is called the rejector and the object 9 D E V, assigned to an object D E 9, is called the rejection of D. Since V is a subcategory of 9,the universal problem corresponding to a reflexive subcategory is easily represented. Let C E V and D E ~ . There exists a morphism f :D + W D in 9 induced by the natural transformation Id9 + 8%.If g : D + C is another morphism in 9, then there exists exactly one morphism h in the subcategory V which makes the diagram f

D-gD

C

commutative. Dual to the notions defined above, a subcategory d : V --+ 9 is called a corejexive subcategory, if d has a right adjoint functor W : 9 + V. Correspondingly, W is called the coreJlector and 9 D the coreflection of the object D E 9. We give some examples for which the reader who is familiar with the corresponding fields will easily verify that they define reflexive or coreflexive subcategories. Some of the examples will be dealt with in more detail in later sections. Reflexive subcategories include (1) the full subcategory of the topological T,-spaces ( i = 0, 1,2, 3) in Top, (2) the full subcategory of the regular spaces in Top, (3) the full subcategory of the totally disconnected spaces in Top, (4) the full subcategory of the compact hausdorff spaces in the full subcategory of the normal h a u s d o d spaces of Top, (5) the full subcategory of the torsion free groups in Ab, (6) Ab in Gr,and (7) the full subcategory of the commutative, associative, unitary rings in Ri. T h e full subcategory of the torsion groups in Ab gives an example of a coreflexive subcategory. Other examples for coreflexive subcategories are the full subcategory of locally connected spaces in Top, and the full subcategory of locally arcwise connected spaces in Top.

LEMMA.Let %? be a full, rejexive subcategory of the category 9 with rejector W.Then the restriction of W to the subcategory V is isomorphic t o Idy.

Proof. Since

%7 is a full subcategory, we get for each C E V that the morphism 1, : C --f C is a universal solution for the universal problem defined by € : 5F --+ 9. By the uniqueness of the universal solution W C g C is natural in C for all C E V.

2.4

REFLEXIVE SUBCATEGORIES

75

I n the case of a reflexive subcategory we have a simple presentation of the universal problem defined by the adjoint functors; thus it is interesting to know when a pair of adjoint functors induces a reflexive subcategory. T h e following theorem gives a sufficient condition.

THEOREM 1. Let the functor 9 : V -+ 9 be left adjoint to the functor B : 9 -+ V and let 59 be injective on the objects. Then B(9) is a reflexive subcategory of V with reflector 5 9 9 . Proof. T h e image of g is a subcategory of V be a remark at the beginning of Section 1.8. We define factorizations of the functors by the following commutative diagram of categories: 9’ v-v‘

91 sf

18

9-vB

where V’ = Y ( 9 ) . By Section 2.1, Corollary 4 we have that 59 : Mor&F--, -)

--*

Morw(%F--,

9-)

is injective. Thus, B‘ : Mor9(9--, -) 4Morw.(S’S-, S’-) is a natural isomorphism by the definition of V‘. We get Morw,(%‘S-, 59’-)

MorB(S-,

-) g Morw(-, 9-) g Mary(--, &’3’-)

Since each object in V’ may uniquely be represented as B’D, and since B’ is full, we get Morw.(W-, -) MOTyp(-, 8-).F and 9 9 coincide up to the embedding of V’ into V. Let V’ be a reflexive subcategory of V with reflector B. PROPOSITION. For all A E V‘ the morphism f : A -+ W A defined by the corresponding universal problem is a section in V.

-

Proof. Let d : V’ -+ W be the embedding. By Section 2.1, Theorem 1 08 8Y 8Y’A we have (& +dRd 8)= id,, thus ( A W A -A) = 1, for all A E C’. Observe t h a t f i s a morphism in C, whereas &!PA is even in V .

THEOREM 2. Let d : % 9 be a full, reflexive subcategory. If for each C E V also each D E 9 with C D in 9 is an object in V, then d is a monadic functor. -+

76

2.

ADJOINT FUNCTORS AND LIMITS

Proof. Let % = 8W and W be the reflector to b, then r(D) : D +8WD is the universal solution of the universal problem defined by 8. Let 6 : %D ---+ D be a 9 morphism, such that

is commutative. Then c(D)&(D) = c(D). Since d is full, we get r(D)S = &( f ) with f :W D --+ WD. By the universal property of E(D) and the commutativity of D

4D)

89D

89D

we get f = lgD, thus r(D)6 = l H D . This proves that r(D)+ # D is an isomorphism and D E V. Furthermore, because (YWD)(We(D))= lAD= (WS)(Wr(D)), we also have YWD = 9 6 , thus p ( D ) = S 6 . This implies that 2 X D

H8

dD)I

2D-D

8

X D

l8

is commutative, and (D, 6 ) is an X algebra. If D E V , then there exists exactly one 6 : #D + D with 6c(D) = 1, , because c(D) is a universal solution. Let f : D + D’ be a morphism and D, D’ E V. Let (D, 6) and (D’, 6’) be the corresponding %-algebras. Then

is commutative, thusf is a morphism of #-algebras. Hence 2’: V is an isomorphism of categories.

+ 9#

2.5 2.5

LIMITS AND COLIMITS

77

Limits and Colimits

Let d be a diagram scheme, V a category and Funct(d, V) be the diagram category introduced in Section 1.8. We define a functor X :V + Funct(x2, V) byX(C)(A) = C, X ( C ) ( f ) = lCand%(g)(A) = g for all C E V, A E d,f E d,and g E V, and we call X the constant functor. In the inversely connected category Y>(Funct(d, V), V), with the connection Mory(C, 9) = Mor,(XC, 9), the functor Xdefines a universal problem for each diagram 9E Funct(d, V). We want to find an object U ( 9 ) in V and a morphism psF : U ( 9 )+ S, such that to each morphism cp : C -+ F there is exactly one morphism v* : C -+ U ( 9 )with p.Fcp* = y. If d is the empty category, then Funct(x2, V) consists of one object and one morphism. X maps all objects of V to the object of Funct(d, W ) and all morphisms to the morphism of Funct(d, V). Since Mor,(XC, 9) has one element, the object U ( 9 ) must satisfy the condition that from each object C E V there is exactly one morphism into U ( 3 ) .Thus, U ( 9 ) is a final object. We formulate the universal problem more explicitly. First, a morphism v E Mory(C, 9)= Mor,(YC, 9)is a family of morphisms ?(A) : C + F A , such that for each morphism f : A -+ A’ in d the diagram

is commutative. In particular psF is such a family of morphisms to make the corresponding diagrams commutative. This family of morphisms has to have the property that to each family tp E Mor,(%C, 9) there is exactly one morphism v* : C + U ( 9 ) such that the diagram

pF(A): U ( 9 )-&A,

C

is commutative for all A E d. If there is a universal solution for the universal problem defined by 9, then this universal solution is called the limit of the diagram 9and

78

2.

ADJOINT FUNCTORS AND LIMITS

is denoted by liip 9.T h e morphisms pF(A) : lim 9-+ P A are called projections and are denoted by p , = pg(A). If the diagram 9is given as a set of objects Ci and of morphisms in U, then we often write lim C, instead of 1 $ 9 . Since the notions introduced here are very important, we also define the dual notion explicitly. T h e constant functor X defines a directly connected category -t’,(Funct(d, U),U) with the connection Mory(F, C) = Mor,(F, X C ) . T h e universal problem which belongs to a diagram 9 may be explicitly expressed in the following way. Each morphism cp E M o r y ( P , C) = M o r t ( 9 , X C ) is a family of morphisms y ( A ) : F A -,C, such that to each morphism f : A --f A‘ in d the diagram

FA’ is commutative. Then in particular pF is such a family of morphisms

pF(A) : 9 A + U(P), which makes the corresponding diagrams commutative. We require that this family of morphisms has the property that to each cp E M o r t ( 9 , X C ) there is exactly one morphism rp* : U ( 9 )--t C such that for all A E d the diagram F A

-

P ~ ( A )

U(S)

1..

C is commutative. If there is a universal solution for the universal problem defined by S, then this solution is called the colimit of the diagram 9 and is denoted bym;1 9. T h e morphisms pS(A) : 9 A -+ lkm 9 are called injections. If the diagram P is given as a set of objects C,and a set of morphisms in U, then we often write lim C,instead of 1 5 9. If there is a limit [colima for each 9~ F u n c t ( d , U), then V is called a category with d - l i m i t s [at-colimits]. If there are limits [colimits] in U for all diagrams 9 over all diagram schemes d , then V is called complete [cocomplete]. Correspondingly, we define a finitely complete [respectively, cocomplete] category, if there are limits [colimits] in V for all diagrams over finite diagram schemes d.

2.5

LIMITS AND COLIMITS

79

LEMMA1. Let 9: d -+ V be a diagram. If the limit or colimit exists, then it, respectively, is uniquely determined up to an isomorphism. Proof. Limits and colimits are unique up to an isomorphism because they are a universal solution.

LEMMA 2. A category V is a category with &-limits [&-colimits] $ and only if the constant functor X :V -+ Funct(&, U ) has a r k h t adjoint [left adjoint] functor. Proof. Since the limits are universal solutions, the lemma is implied by Section 2.2, Theorem 1. T h e explicit formulation of the universal problem defining a limit allows us also to define a limit for functors .F : A? -+ V with an arbitrary category B. But limits of these large diagrams will not always exist, even if V is complete. Compare the examples at the end of this section. Now we want to collect all diagrams over a category V (not only those with a fixed diagram scheme) to a category. We have two interesting possibilities for this. T h e category to be constructed will be called the large diagram category, and we denote it by 9g(V). T h e objects of Q ( V ) are pairs (d, 9), where d is a diagram scheme and .F : & -+ V is a diagram. T h e morphisms between two objects (&,and 9) (&’,St) are pairs (9, p’), where 9 : d -+ d’is a functor and q~ : 9-+ 9’9 is a natural transformation. Now, if morphisms (9, q ~ ): (d, 9) -+ (d’, 9’)and (9’, p”) : (d’, 9‘ -+ ) (d”, 9”) are given, then let the (p’”3)q~). With this composition of these two morphisms be (g‘9, definition, IDS(%) forms a category. We also construct another large diagram category ag’(V) with the same objects as in as(%?), in which, however, a morphism from (d, 9) to (d’,9’)is a pair (9, p’) with a functor 9 : & -+ &’ and a natural transformation p’ : 9’3 -+ 9. T h e composition in ag‘(V) is ( 3 ’ 9

F’)P, F) = (3’9, F(F’m

For each diagram scheme d , the category Funct(M’,V) is a subwith the application 9tt (&, 9) and ? I+ (Id,, ?). category of as(%‘) Similarly, Funct(&, V)O is a subcategory of Bg’(V). Both subcategories are not full because there may be other endofunctors of & than Id,. Let 0 be a discrete category with only one object. T h e composition of the constant functor X : V Funct(0, V) with the embedding Funct(0, V) -+ IDS(%) will also be called the constant functor and will be denoted by R : V -+ ag(V). Similarly, we get a constant functor --f

R

: %?o -+ 9g’(V).

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PROPOSITION 1. The category V is cocomplete if and only functor R :V -B as(%) has a left adjoint functor.

g the constant

Proof. Let us denote MorDocwp,((d, 9), RC) by Mor((d, S), RC). R has a left adjoint functor if and only if M o r ( ( d , S ) , 52-) is representable for all (d, 9) (Section 1.15, Lemma 4). Let (8,cp) E Mor((d, S), RC), then 2'2 : d + 0 is uniquely determined, and we have a natural transformation cp :S + Z,C, where Z, : V + Funct(d, V ) is the constant functor. The functor corresponding to RC composed with 8 assigns to each object in d the object C E V and to each morphism in a? the morphism l c E V . Thus Mor((d, F),RC) E Mor,(F, .X,C). It is easy to verify that this isomorphism is natural in C; Mor((d, 9), R-) Mort(Fy Z,-). The functor Mort(*, Z,-) is representable for all (d, S)if and only if V is cocomplete (Lemma 2). PROPOSITION 2. The category V is complete g and only junctor R : Vo + Iog'(V) has a left adjoint functor.

if the constant

Proof. This proposition is implied by Proposition 1 if one replaces V by Vo. In fact, E Bg'(V). In particular, the following notations make sense. Let 9 : d -+V and be functors, and let cp :9 + 8 be a natural transformation. Then let 1% cp = l$(Id, ,cp) and l@ cp = lim(Id, ,cp), where 1Lm and $m denote the left adjoint functor for R wi& values in V of Proposition 1 and Proposition 2 respectively (also in the case of Proposition 2). We write also $9 : d + V

lim v : lim 9-t lim 9 + t +

and

lim q~ : 1 i m P - t lim 9 C

t

t

Let 9 : d + V , 8 :AY + V, and i@ : a? -B L47 be functors, such that the diagram

is commutative. We assume that here both d and AY are small categories. Then we define 1 5 i@ : 1% F + 1% 8 and l@ i@ : 1 2 F -+i@ $9 by l i z 3P ' = 1$(X, id,) and I@ X = l i p ( X , id,) respectively.

2.6

SPECIAL LIMITS AND COLIMITS

81

Now we want to investigate when a small category d is complete. Let Mor,(A, B ) be a morphism set with more than one element. Let I be a set which has larger cardinality than the set of morphism of A . Finally, let B, = C with B, = B for all i E I . Then the cardinality of Mor,(A, C) is larger than the cardinality of the set of all morphisms of d.Thus each morphism set Mor,(A, B ) can have at most one element. A similar argument holds for a cocomplete small category. Now let us define A < B if and only if Mor,(A, B ) # IZ(, then this is a reflexive and transitive relation on the set of objects of d .Such a category is also called pre-ordered set. Often a limit is also called an inverse limit, projective limit, infimum, or left root. Correspondingly, a colimit is often called a direct limit, inductive limit, supremum, or right root. We shall use these notations with a somewhat different meaning.

niE,

2.6 Special Limits and Colimits I n this section we shall investigate special diagram schemes d and the limits and colimits they define. Some of these examples are already known from Chapter 1. Let d be the category

that is, a category with two objects A and B and four morphisms l,, l B , f : A + B, and g : A + B ; let .F:d + V be a covariant functor, then l i E F = Ker(Ff, Fg). I n fact, let us recall the explicit definition of the limit. A natural transformation v :X C + 9 is a pair of morphisms ?(A) : C - t 9 A and v(B) : C + 9 B , such that 9(f) v(A) = v ( B ) = F ( g ) v(A). This is equivalent to giving a morphism h : C -+F A with the property F ( f ) h = F ( g ) h . T h e difference kernel of (Ff, 9 g ) is a morphism i : Ker(9f, 9 g ) + .FA with the property that to each morphism h : C + 9 A with this property, there is exactly one morphism h’ : C + Ker(Ff, F g ) with h = ih’. This is exactly the definition of the limit of F. Here i is the projection. Dually, lim F = Cok(Ff, 9 g ) . + Let d be a discrete category, which we may consider as a set I by Section 1.1. Then a diagram 9over d is a family of objects {Ci}iE,in V. T h e conditions for the limit l i p F of F coincide with the conditions for the product Ciof the objects C, . T h e projections of the product into each single factor coincide with the projections of the limit into the objects 9 ( i )= C, . Correspondingly, the colimit of 9 is the coproduct of the C, .

niE,

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Another important example of a special limit is defined by the diagram scheme

I that is, by a small category a2 with three objects A , B, C, and five morphisms 1, , I , , l c ,f: A + C , and g : B --+ C. A natural transformation : X D -+ 9for an object D E V and a diagram 9is completely described by the specification of two morphisms h : D + .FA and k : D + .FB with F(f )h = 9 ( g ) k . T h e limit of .F consists of an object F A x FB 9C

and two morphisms PA

:F A

x 9B-t F A

and

sc

PB

: 2FA x 2FB-+FB 9FC

with S ( f ) p A = F ( g ) p s , such that to each triple ( D , h, k) with .F(f )h = .F(g)k there is exactly one morphism l:D+FA x F B .%C

such that the diagram

is commutative. This limit will be called fiber product of .FA and .FB over 912. Other names are Cartesian square and pullback. Let &’ be dual to the diagram used for the definition of the fiber

product; thus let d be of the form

1

2.6

SPECIAL LIMITS AND COLIMITS

83

Let 9 be a diagram over d in %. The colimit 1Lm 9 will be called a cojiber product. Other names are cocartesian square, pushout, fiber sum, and amalgamated sum.

PROPOSITION 1, Let ‘3be a category with jinite products. V? has dzyerence kernels if and only if% hasjiber products. Proof. Let % have difference kernels. In the diagram

A

let (A x B, p , ,p,) be a product of A and B, and let (K, q) be a difference kernel of ( f p , ,gp,). Furthermore, let qa = pAqand q, = pBq. Then the diagram is commutative, except for the pair of morphisms ( f p A ,gps). We claim that (K, qA , q,) is a fiber product of A and B over C. I n fact we have f q , = gq, . If h : D + A and k : D + B is a pair of morphisms of % with f h = gk, then there is exactly one morphism (h, k) :D +A x B with h = p,(h, k ) and k = pB(h,k ) . Hence, fp,(h, k ) = gp,(h, k ) . So there exists exactly one morphism 1 : D --t K with ql = ( h , k ) , and we have q,l = h and q,l = k . The diagram extended by h : D + A and k : D --t B becomes commutative if we add 1 : D + K (except for f p , ,gp,); this implies that 1 is uniquely determined. Let %? have fiber products. I n the commutative diagram

KPAA pE

1

l(f..)

B%BXB

le B x B be the product of B with itself, A , th diagonal, ( f , g ) the morphism uniquely determined by two morphisms f : A + B and g : A + B, and let (K, p a , p,) be a fiber product. We claim that (K, PA) is a difference kernel of the pair of morphisms ( f , g ) . (Distinguish between the pair of morphisms ( f , g ) and the morphism ( f , g ) ) . Now let q1 : B x B + B and q2 : B x B ---t B be the projections of the product.

84

2.

ADJOINT FUNCTORS AND LIMITS

Then we have ( f, g) PA= dBp,

fPA

, thus

= ql(f, g) PA = qlABPB =

lBPB

= qZAB@B = q2(f, g) P A = g P A

Let h : D --t A with fh = gh be given. Then fh : D B and qld, fh = lBfh = qJ.(f,g)handqdBfh = lBfh = lBgg = q2(f,g)h,thusdBfh= (f,g)h. Consequently, there exists a unique morphism k : D ---t K with p,k = h and pBk = fh(= gh). But this is the condition for a difference kernel. Difference kernels may also be represented in a different form as fiber products. This will be shown by the following corollary. --f

COROLLARY 1. Let f, g : A diagram

--t

K-A

B be morphisms in V. The commutative P

p)

A’~A,~’-AXB is a fiber product if and only if (K,p) is a dtrerence kernel of the pair ( f, g). Proof. The hypothesis that both projections K -+ A of the fiber product coincide is no restriction, since if h, k : C A are two morphisms with ( lA,f )h = ( l A,g)k, then by composition with the projection A x B -+ A we get the equations h = k andfh = gh. Thus the claim follows directly from the definition of the fiber product and the difference kernel, --j

LEMMA 1. Let V have fiber products and a final object. Then V is a category with fznite products. Proof. Let E be a final object in V. Let A and B be objects in V. Then there is exactly one morphism A E and exactly one morphism B +.E. Assume that the commutative diagram ---f

K-A

B-E

is a fiber product. Then K is a product of A and B. The requirement that the square be commutative is vacuous because there is only one morphism from each object into E.

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SPECIAL LIMITS AND COLIMITS

PROPOSITION 2. Let g be a category with (finite) products and dzflerence kernels. Then $9 is (jinitely) complete. Proof. Let d be a diagram scheme and 9 : &-+%? be a diagram. Let P = n A E , 9 A . Let Q = n l e , 9 R ( f ) where R ( f ) is the range off. For each object 9 R (f ), we get two morphisms from P into 9 R (f ), namely for f : A --t A’ we get the projection p,, : P + %A‘ and the morphism 9( f ) p , : P + F A -+ FA’. This defines two morphisms p : P + Q and q : P - Q . Let K = Ker(p, q). Let cp : X C 9 be a natural transformation. Then for all A E A? there are morphisms ?(A): C -+ 9 A with the property that ---f

FA‘ is commutative for allf

E

&. T h u s the compositions

are equal, that is, there is exactly one morphism cp* : C -,K such that

is commutative. Thus, K is a limit of 9. COROLLARY 2. The categories S and Top are complete and cocomplete. Proof. By Sections 1.9 and 1.11 both categories have difference kernels and cokernels, products and coproducts. Proposition 2 and the dual of Proposition 2 give the result.

3. A category with fiber products and a final object is finitely COROLLARY complete. T h e proof is implied by Proposition 1, Lemma 1, and Proposition 2.

86

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ADJOINT FUNCTORS AND LIMITS

COROLLARY 4. Let V be a complete category and let 9 :V + 9 be a functor which preserves dayerence kernels and products. Then 9 preserves limits. Proof. By Proposition 2, a limit is composed of two products and a difference kernel. These products and difference kernels in ‘X are transferred by @ into corresponding products and difference kernels in 9. Thus they also form a limit in 9 of the diagram which has been transferred by 9 into 93.

A functor preserving limits [colimits] is called continuous [cocontinuous]. I n particular, such a functor preserves final and initial objects as limits and colimits respectively of empty diagrams. A special fiber product is the kernel pair of a morphism. Let p : B + C be a morphism. An ordered pair of morphisms

(f,,: A -+ B, fi : A -+ B ) is called a kernel pair of p if (1) p f , = p f , and ( 2 ) for each ordered pair (h, : X + B, h, : X - + B )

withph, = ph, , there is exactly one morphismg : X +-A with h, = f o g and h, = f , g : X

A

,

JO

B

-

C

P

fi

( f , ,f , ) is a kernel pair of p if and only if A is a fiber product of B over C with itself:

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SPECIAL LIMITS AND COLIMITS

If there are fiber products in %, then there are also kernel pairs of arbitrary morphisms in %?.

LEMMA2. g : A kernel pair of g.

-+

B is a monomorphism if and only if (1,

, lA) is a

Proof. Let h, , h, : X + A be given with gh, = gh, . I n such a case g is a monomorphism if and only if we always have h, = h, , This is true ifandonlyifthereisamorphismf:X-tAwithI,f =h,andl,f = h,.

COROLLARY 5. If a functor preserves kernel pairs, then it preserves monomorphisms. LEMMA3. I n the commutative diagram f

K

f'

g'

A-B-C

A' +B' +C'

let the right square be ajiberproduct. ( A ,f , a ) is ajiberproduct of B and A' over B' if and only if ( A ,gf, a ) is a jiber product of C and A' over C'. Proof. Let ( A ,gf,a) be a fiber product. Let h : D -+ B and k : D A' be morphisms with bh = f 'k. Then we get for gh : D + C and for k : D + A' the equation cgh = g'f'k. Thus there is exactly one x : D +A with gfx = gh and ax == k. We show f x = h. In fact, then ( A ,f , a ) is a fiber product of B and A' over B'. We have gh = gh and bh = f 'k. Furthermore, we havegfx = gh and bfx = f 'ax = f 'k. Since the square is a fiber product, the equation fx = h is implied by the uniqueness of the factorization. Let ( A ,f, u ) be a fiber product. Let h : D --t C and k : D + A' be morphisms with ch = g'f'k. Because of ch = g'( f ' k ) , there is exactly one x : D -+ B with bx = f 'k and gx = h. Because of bx = f ' k , there is exactly one y : D -+ A with fy = x and ay = k. Then the uniqueness of y with gjy = h and ay = k follows trivially. --f

A small category d is called filtered if:

(1) for any two objects A, B Ed there is always an object C E together with morphisms A -+ C and B + C, and for any two morphismsf, g : A + B there is always a morphism (2) h : B + C with hf = hg.

~

88

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ADJOINT FUNCTORS AND LIMITS

A small category d is called directed if it is filtered and if each morphism set Mord(A, B) has at most one element. Let 9: d ---t V be a covariant functor. If a? is filtered, then & l m F is called a jiltered colimit. If d is directed, then 1% 2F is called a direct limit. Let 9:d o---t V be a covariant functor. If d is filtered, then l@ 9 is called a jiltered limit. If d is directed, then l i p 2F is called an inverse limit. These special limits and colimits will be very important for abelian categories discussed in Section 4.7 Now we give some examples of finitely complete categories, without proving this property in each particular case: the categories of finite sets, of finite groups, and of unitary noetherian modules over a unitary associative ring. Furthermore, we observe that in S, Gr, Ab, and .Mod each subobject appears as a difference kernel. I n Hd exactly the closed subspaces are difference kernels, in Top all subspaces are difference kernels. This may be proved easily with the dual of the following lemma.

LEMMA 4. Let V be a category with kernel pairs and daference cokernels. (a) f is a dzference cokernel i f and only i f f is a daference cokernel of its kernel pair. (b) h, , h, : A ---t B is a kernel pair if and on& i f it is a kernel pair of its daference cokernel.

Proof. We use the diagram X

C

D (a) Let f be a difference cokernel of ( g o ,g,), and let (h, ,h,) be a kernel pair off. If kh, = kh, , then kg, = kg,; thus there is exactly o n e y with yf = k. (b) Let (h, ,h,) be a kernel pair of k and let f be a difference cokernel of (h, ,hl). Then there is exactly one y with k = y f . I f g o ,g, are given withfg, = fgl ,then kg, = kg, ,thus there is exactly one x with hix = g, for i = 0, 1.

2.7

89

DIAGRAM CATEGORIES

2.7 Diagram Categories I n this section we discuss mainly preservation properties of adjoint functors, limits, and colimits. For this purpose, we need assertions on the behavior of limits and colimits in diagram categories.

THEOREM 1. Let d be a diagram scheme and U be a (finitely) complete category. Then F u n c t ( d , U ) is (jinitely) complete, and the limits of functors in F u n c t ( d , U ) are formed argumentwise. Proof. Let A9 be another diagram scheme. Let A? : V -+ Funct(3, U ) and X ’ : Funct(.d, U ) -+ Funct(g, F u n c t ( d , U ) ) be constant functors. Let 2Y E F u n c t ( d , U ) and E F u n c t ( d , Funct(A?, U ) ) . Let A ?.% be the composition of functors, and let q : A?&‘ -+ ’3 be a natural transformation. Then to each v(A)E M o r ( X H ( A ) , B ( A ) )there is a v‘(A) E Mor(H(A), lcm(3(A))) such that the following diagram is commutative: Mor(XZ(A), 9 ( A ) )

N

l+

M o r ( J P ( A ) limW(f))) ,

/Mor(XJP(A).W))

Mor(X&‘(A),

%(A’))

N

t

Mor(Z(A), lim(S(A’)))

t-

Mor(JP(f),lim(S(A 9))

fMor(.TX’(f),9(A‘))

Mor(XZ(A’), S(A’))

Mor(Z(A), lim(g(A)))

N -

t

Mor(Z(A’), lim(%(A’))) t

where f : A --+ A’. v(A’)XH(f ) = 9(f ) v ( A )implies v’(A’)2Y(f ) = l@(’3( f )) rp’(A), that is, v’ : 2 -+ lip(9(-)) is a natural transformation. So we have M o r l ( X X , 9) Mor,(H, I@ ’3). We define Funct(d, X ) : Funct(d, U) + Funct(d, Funct(22, U)) by Funct(.d, X ) ( X )= X 2 and F u n c t ( d , X ) ( p ) logously

=

A?p and ana-

Funct(92, lim) : Funct(d, Funct(9, U ) )4Funct(d, V) c Then Funct(d, X ) is left adjoint to Funct(a2, I@). F u n c t ( d , X ) with the isomorphism

If we compose

Funct(.r/, Funct(22, %‘))g Funct(g, Funct(d, U ) ) we get the functor X ’ , which has a lcft adjoint functor lim’ : Funct(22, Funct(d, 9)) + Funct(d, ‘$7) t

2.

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ADJOINT FUNCTORS AND LIMITS

Here l@’(%’)(A) = $(%’(A)), which means that the limit is formed argumentwise. Observe that we identified the functor B E Funct(d, Funct(99, V))

with the corresponding functor in Funct(@, Funct(d, U)). Dualization of d and V implies the dual assertion that, with %, Funct(cc4, V) is also (finitely) cocomplete and that the colimits are formed argumentwise. For this purpose, use Funct(d, U ) E Funct(do, Y0)O of Section 1.5.

THEOREM 2. Let d be a diagram scheme, 9 : d -+ U a diagram in g, and C E V. If l i p 9 or l i z 9 exist, then there are, respectively, isomorphisms l@ Mor(C, 9)g Mor(C, lim 9) c

lirn Mor(9, C ) g Mor(1im 9,C) t

+

which are natural in 9and C. Proof. Let & = Funct(d, U), g2= Funct(d, S), 9E and X E S.Then Morg2(XX, Mory(9-, C ) )

,

C E V,

Mors(X, Morgl(9, ZC))

natural in 9, C, and X. In fact, let f E Mor,,(XX, Mory(9-, C)), then f is uniquely determined by f ( A ) ( x ) : $A + C for all x E X and natural in A E d.We assign g ( x ) ( A )= f ( A ) ( x ): 9 A + C to f . Then g E Mor,(X, Morgl(F, X C ) ) . This application is bijective and natural in 9, C, and X. Thus, by changing to the functor which is adjoint to X we obtain Mors(X, lim Mory(9, C ) ) c

Mors(X, MorV(l5 9,C ) )

and thus l@ Mor,(9, C) Mory(l$ 9, C).We obtain the other assertion dually. Here again the consideration preceeding Section 1.5 on the generalization of notions in S to arbitrary categories with representable functors are valid. In particular, this theorem generalizes the remark at the end of Section 1.1 1 .

COROLLARY 1. Let 9 : cc4 -+ %? be a diagram. Let C E W. Then the limit of the diagram h C S : s2 -+ S is the set Mor,(XC, g).

2.7

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DIAGRAM CATEGORIES

Proof. In the proof of Theorem 2 there is an isomorphism Mors(X, lirn Morw(9,C))

Mors(X, Morf(9, X C ) )

c

which implies lim M o r w ( 9 , C) g M o r f ( S , Z C ) . T h e assertion of the corollary is dual. Observe that we do not need the existence of 1% S for this proof.

THEOREM 3. Left adjoint functors preserve colimits; right adjoint functors preserve limits.

Proof. Let F : U + 9 be left adjoint to $9 : 9 -+ V. Then we have for a diagram S : d -+ 9 and an object C E U Mor(C, 9 lirn 8) Mor(SC, lirn X) t c N lirn 4-

Mor(C, 9 8 )

lirn Mor(9C, #) c

Mor(C, lirn 9s) c

This implies 9 I@ S g li@ $92One . gets the second assertion dually.

LEMMA1. Let 9 : d :< g -+ U be a diagram o v m the diagram scheme d x 9. Let there be a limit of 9 ( A , -) : 9 -+ % f o r all A ~ dThere . is a limit of 9 : dx -+ U i f and only if there is a limit of 9 : d -+ Funct(g, U ) . If these limits exist, then we have lirn lim 92 lim 9

dTd

t t

&fa

Proof. T o explain over which diagram the limit is to be formed, we wrote the corresponding diagram schemes under the limits. Corresponding functors in F u n c t ( d x B, U), F u n c t ( d , Funct(9, U)), or Funct(L@’,F u n c t ( d , 9))will be denoted by no prime, one prime, or a double prime respectively. Since 1&1~(9(A,-)) exists for all A E d, lima(F”) also exists. Then we have c Morw(C,l i m ( 9 ) )

Mor,(.X&,C,

9) g Mor,((.XBXdC)”, 9”)

d&? N Morf(.XdC, lirn 9”) = t

d

Mory(C, lirn lirn 9”) C

d

t

d

natural in C E V. Here Xdxd: U -+ F u n c t ( d x &?, U), .Xd : ‘3+ F u n c t ( d , U ) , and X, : F u n c t ( d , U ) -+ Funct(B, F u n c t ( d , U)) are constant functors.

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ADJOINT FUNCTORS AND LIMITS

COROLLARY 2. Limits commute with limits and colimits commute with colimits. Proof. Obviously lim 9g lim .F c

4d X a

I X d

thus, lim lim 9g lim lirn 9 4 - +

&

I

c c

a d

COROLLARY 3. The constant functor 3": V limits and colimits.

-+

Funct(d, V) preserves

Proof. We have Mor(9, X lim 9)gg lirn Mor(9, lim g) t t + I d a N -

lirn lim Mor(.F,%) c + a &

Mor(9, lim X 9 ) c a

where 9 : d -+ V and $2 : 9 -+ V.

LEMMA 2. Let d be a small category, V an arbitrary category, 9, $2 : d --t V functors, and v : 9-+ $2 a natural transformation. If F A is a monomorphismfor all A E&', then is a monomorphism in Funct(d, V). Let V be finitely complete and cp be a monomorphism, then F A is a monomorphism for all A E &'. Proof. Two natural transformations $ and p coincide if and only if they coincide pointwise ($A = PA). Thus the first assertion is clear. For the second assertion, we consider the commutative diagram in Funct(&', V) 9 x 9

1.

id91

9

L

9

which is a fiber product by Section 2.6, Lemma 2. By Theorem 1, this

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DIAGRAM CATEGORIES

93

is a fiber product argumentwise for each A E d . Then again by Section is a monomorphism for all A ~ d . 2.6, Lemma 2 we get that

COROLLARY 4. Let d be a diagram scheme and U be aJinitely complete, locally small category. Then F u n c t ( d , U ) is locally small. Proof. By Lemma 2, monomorphisms in F u n c t ( d , U ) are formed argumentwise. Similarly, the equivalence of monomorphisms holds argumentwise. In fact, if two natural monomorphisms in F u n c t ( d , V )are , the family of uniquely equivalent for each argument A E ~ then determined isomorphisms of the equivalences defines a natural isomorphism which induces the equivalence between the two given natural monomorphisms. Now since d is a small category and since U is locally small, there can only be a set of subobjects for an object in F u n c t ( d , U).

COROLLARY 5. Let

be a fiber product and let f be a monomorphism. Then p , is also a monomorphism.

Proof. T h e commutative diagram

is a morphism between two fiber products. Since f, l c , and 1, are monomorphisms, the corresponding natural transformation is a monomorphism, thus by Corollary 2 and Section 2.6, Corollary 5 the morphism p , is also.

2.

94

ADJOINT FUNCTORS AND LIMITS

LEMMA 3. ( a ) Right adjoint functors preserve monomorphisms. Left adjoint functors preserve epimorphisms. (b) Let 9, 3 : d -+ V be diagrams in V and let pl : F + 3 be a morphism of diagrams with monomorphisms plA : F A -+ 9 A . I f ljm pl : l $ F -+ l@ 3 exists, then lim rp is a monomorphism.

Proof. (a) is implied by Theorem 3 and Section 2.6, Corollary 5 . (b) is implied by Lemma 2, Corollary 2, and Section 2.6, Lemma 2.

THEOREM 4 (Kan). Let d and 9 be small categories and let V be a cocomplete category. Let F : a? be a functor. Then Funct(F, V ) : Funct(d, $7) + Funct(9, $7) has a left adjoint functor. -+

Proof. First we introduce the following small category. Let A E&. Then define [F, A] with the objects (B,f ) with B E and f : F B -+ A in d.A morphism in [F, A] is a triple (f, f ’, u ) : ( B , f )-+(B’,f’)with u : B -P B’ and f ’ F u = f . A functor Y ( A ): [S, A] + @ is defined by Y ( A ) ( Bf, ) = B and V ( A ) (f,f ’, u ) = u. Let g : A -+ A’ be given. We define a functor [S, g ] : [F, A] + [F, A’] by [*, g l ( B , f ) = (4g f ) and 19, g1( f , f ‘,u ) = ( g f , gf’, u). Thus in particular, V ( A )= V ( A ’ ) [ Fg,] . Define a functor 59 : Funct(9, V) -+ F u n c t ( d , V) by 59(Z‘)(A)= 1 2Z‘W(A),3 ( 2 ) ( g ) = l i m [ 9 , g ] : lim #V(A’), and + Z‘W(A)+ lim 4 3 ( a ) ( A )= l$(aV(A)). W< want to show that 9 is left adjoint to Funct(9, V). Let A? E Funct(g, ‘27) and 9 E Funct(&’, V ) be given. We show Morf(9(iF), 2)

If

pl

: 9(Z)-+

Mor,(iF, 2.F)

9 is a natural transformation, then p(SB) : lim S V ( . F B ) -P 9 S B . --f

Since (B, lFB) E [F,F B I , there is an injection i : Z‘B +lim + XV(FB). Set $(B) = pl(FB)i. This defines a family of morphisms

Let h : B

-+ B’ be a morphism in B. Then we get [S, S h ] : [S,S B ] -+ [F,FB’], thus @[F, S h ] : lim + X V ( S B ) -+ 1% Z‘V(.FB’). Since

2.7

DIAGRAM CATEGORIES

95

is a natural transformation and because of the properties of the colimit, the diagram

-

is commutative and thus I) is a natural transformation. Let I) : 2 99be given. Let A E d.To each pair (B,f) E [F, A] we get a morphism

If (f,f’, u ) E [F, A], then

is commutative; thus there is exactly one morphism v(A): 1% 2 V ( A )-+ 9 ( A )such that the diagram

is commutative. Because of the properties of the colimit, the following diagram with g : A -+A’ is also commutative

96

2.

ADJOINT FUNCTORS AND LIMITS

Thus 37 is a atural transformation. Because of th uniq mess of v the application v tt t,h t-+ tp is the identity. Furthermore, o6e checks easily that t,h I-,9) I-+ t,h is the identity. Thus, Mor(9(3Ea), 9) M o r ( 2 , 99). The given applications imply that this isomorphism is natural in & and 9. This proves the theorem.

COROLLARY 6. Let U be cocomplete and 9 : a + d be a functor of small categories. Then Funct(9, U) : Funct(d, U)+ Funct(a, U) preserves limits and colimits. Proof. Funct(S, U)is a right adjoint functor; consequently it preserves limits. Since in Funct(d, V) and in Funct(d?, U ) there exist colimits that are formed argumentwise (Theorem 2), we get for a diagram 3? : 9 -+ Funct(d, U) lim Funct(9, %?) Z ( B ) = lirn H R ( B ) = Funct(9, U ) lim 4 + -+ S ( B )

COROLLARY 7. Let d and 9Y be small categories and U a complete category. Let F : d? -,d be a functor. Then Funct(R, U ) : Funct(d, U )+ Funct(g, %?)

has a right adjoint functor. Proof.

Dualize d,9,and V.

PROPOSITION 1 . Let A? and 9Y be small categories and V be a n arbitrary category. Let F : d? d be a functor, which has a r k h t adjoint functor. Then Funct(F, U) : Funct(d, U)+ Funct(B, U) has a left adjoint functor. --f

Proof, Let B : d --f 9Y be right adjoint to S and let Q, : Id, 4939 and Y : 9 9 ---+ Id, with (BY)(Q,9)= id, and (Y9)($@) = idF be given.

2.8

CONSTRUCTIONS WITH LIMITS

97

Then we have Funct(@,U) : Funct(Id9, U)--+ Funct(’39, U ) and Funct(Y, Y ) : Funct(9’3, U ) -+ Funct(Idd, U) with

2.8 Constructions with Limits We want to investigate the behavior of the notions intersection and union introduced in Chapter 1 with respect to limits.

PROPOSITION 1. Let W be a category with fiber products. Then % is a category with finite intersections. If V is a category with finite intersections and finite products, then W is finitely complete. Proof. Let f : A fiber product

-+

C and g : B A x C

-+

C be subobjects of C. We form the

BAA

I

By Section 2.7, Corollary 5 , the morphism p , is a monomorphism. Thus, f p , : A >d B -+ C is equivalent to a subobject of C and hence u p to equivalence the intersection of A and B. Given the morphisms f , g : A -+ B . As in Section 2.6, Corollary 1 the difference kernel o f f a n d g is the fiber product of (1, ,f): A --t A x B and (1, ,g) : A -+ A x B. Both morphisms are sections with the retraction p , and hence monomorphisms. This means that we may replace the fiber product by the intersection of the corresponding subobjects. Consequently %‘ has difference kernels. By Section 2.6, Proposition 2, we get that %‘ is finitely complete.

PROPOSITION 2. Let V be a category with j b e r products. Then there exist counterimages in 5f.

98

2.

ADJOINT FUNCTORS AND LIMITS

Proof. Let f :A -+ C be a morphism and g : B -+ C be a subobject of C. Then the fiber product of A and B over C is a counterimage of B under f (up to equivalence of monomorphisms), for p A : A 5 B -+ A is a monomorphism by Section 2.7, Corollary 5. Since we now may interpret counterimages and intersections as limits, we get again the commutativity of counterimages with intersections as in Section 1.13, Theorem 1. In fact, arbitrary intersections are the limit over all occurring monomorphisms.

LEMMA 1. Let V be a category with dzference kernels and intersections. Given f,g , h : A -+ B. Then Ker( f,g ) n Ker(g, h) C Ker( f,h).

Proof. Consider the commutative diagram

Ker(g, h)

A

Then aq = cq' implies f a q = gag = gcq' = hcq' = hag. Thus aq may be factored uniquely through Ker( f,h). Since aq is a monomorphism, we get Ker( f,g ) n Ker(g, h ) C Ker( f,h).

LEMMA 2. Let %? be a category with fiber products and images. Let C C A, D C B, and f : A -+ B be given. Let g : C -+f ( C )be the morphism induced by f. Then we haveg-l( f ( C ) n D ) = C nf -l(D).

Proof. In the diagram C n f - l ( f ( C ) n D ) -f-l(f(C)

1

n D)- f ( C )

nD

1

the outer rectangle is a fiber product because the two inner ones are.

2.8

99

CONSTRUCTIONS WITH LIMITS

Hence C nf-l(f(C) n D)--+f(C) n D

1

1 is a fiber product. Consequently C nf-l(D)

==

C nf-tf(C) nf-l(D)

=

C nf-l(f(C)

n D)= g-l(f(C) n D )

We shall use these lemmas in Chapter 4. I n S the difference cokernel g : B --t C of two morphisms h, , h, : X 4B is a set of equivalence classes in B. I n the corresponding kernel pairf, ,fl : A --f B the set A consists of the pairs of elements in B which are equivalent, or more precisely of the graph R of the equivalence relation in B x B. fo and fi are, respectively, the projections R 3 (a, b) w a E B

and

R 3 (a, b) Hb E B

I n general we define an equivalence relation in a category % as a pair of morphismsf, ,f, : A --t B such that for all X E V, the image of the map (Mor@,fo),

Mor&,fl))

: Morv(X, A )

-

Morv(X, B )

X

MordX, B )

is an equivalence relation for the set Morv(X, B). If (Mor,(X,f,), Mor,(X,f,)) is injective for all X E V, then the equivalence relation is called a monomorphic equivalence relation. If V has products, then we may use a morphism ( fo ,fl) : A ---t B x B instead of the pair fo ,fi : A --t B, because of Morv(X, B ) x Morv(X, B ) g MorV(X, B x B )

T h e pair fa ,fl : A -+B is a monomorphic equivalence relation if and only if it is an equivalence relation and the morphism ( fo ,f,)is a monomorphism. B be a kernel pair of a morphism p : B --t C. Let Let fo ,fl : A P

D'-A

ALB

2.

100

ADJOINT FUNCTORS AND LIMITS

be a fiber product. Then we get a commutative diagram D-A

P

where m is uniquely determined by f o p o : D + B and f l p l : D -+ B with Pfo Po = Pfi Po = Pfo Pl = Pfl Pl * Thus

Furthermore, by Section 2.6, Lemma 3, all quadrangles of the diagram are fiber products. In particular D r A - Bfo Po

and m fl

D:A-B PI

are kernel pairs. The diagrams A

B

PB I 4 and

B-C P

B-C

P

2.8

101

CONSTRUCTIONS WITH LIMITS

determine in a unique way morphisms e : B + A and s : A

foe = f i e

Is

=

--+

A with (4)

and with fos = f i

,

f,s

=lo, and

s2 =

I,,

(5)

This follows from fos2 = fo and fls2 = fl because the lower squares are fiber products. Thus we have obtained a diagram PO + fo -

DAALB L

t

J

L

8

with the properties (I), (3), (4),and (5). Such a diagram is called a groupoid or preequivalence relation. T h e same construction works also if f o ,fl : A + B is not a kernel pair but a monomorphic equivalence relation. I n this case one carries out the construction in S for Moru(X,fo), MorV(x,fJ : M o r ~ ( x4 ,

-

Mor%(X,B )

for all X E V. I n fact, there is a difference cokernel to each equivalence relation in S, namely the set of equivalence classes. Since we consider a monomorphic equivalence relation, Mor,(X, fo),Mor,(X, fl) is a kernel pair for the difference cokernel. Then it is easy to verify that m, , e x , sx depend naturally on X together with the conditions (2), (3), and (4), so that this defines again a groupoid by the Yoneda lemma. T h u s we get part (a) of the following lemma.

LEMMA3. (a) Each kernel pair and each monomorphic equivalence relation is a preequivalence relation. (b) Each preequivalence relation with a monomorphism ( fo ,fl) : A -+ B x B is an equivalence relation. (b) We may identify Mor(X, A ) with the image of (Mor(X,fo), Mor(X,fl)) in Mor(X, B x B ) Mor(X, B ) x Mor(X, B). For each b E Mor(X, B ) the pair (b, b) is in Mor(X, A), since if eb = (b’, b”), then f,(b‘, b”) = foeb = b and fl(b’, b“) = fieb = b, hence eb = (b, b). If (b, b‘) E Mor(X, A), then (b’, b) in Mor(X, A). I n fact, fos(b, b’) = fl(b, b‘) = b’ and fls(b, b‘) = fo(b, b‘) = 6, hence s(b, b‘) = (b’, b). Proof.

102

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ADJOINT FUNCTORS AND LIMITS

Finally, with (b, b’) and (b’, b”) in Mor(X, A ) , also (b, b”) in Mor(X, A). In fact, ((b, b’), (b’,b”)) E Mor(X, D ) g Mor(X, A )

holds because fop,((b, b’), (b‘,b”)) f,p,((b,

x

MOr(X.B)

= fo(b’, b”) =

q,(U, b”)) = f#,

b’)

Mor(X, A )

b‘ and

= b’.

But then fom((b,b‘), (b’, b”)) = fop,((b, b’), (b’, b”)) = fo(b, b’) = b and fim((b, b’), (b‘,b”)) = f l p l ( ( b , b’), (b’, b”)) = fl(b’, b”) = b”, and thus m((b, b’), (b’, b”)) = (b, b”) E Mor(X, A).

LEMMA 4. Let fo ,f l : A -,B be a monomorpkic equivalence relation. For the corresponding groupoid, the following diagrams are commutative: (1 m)

E-D

(ii)

Proof. First we define E, (1, m), and ( m , 1). Let all squares of the commutative diagram be fiber products.

~ 4 1P - 0 ~ ~

Ifo

lPo

P

fl

D-LA-B Pol

Ifo

A-LB

2.8

CONSTRUCTIONS WITH LIMITS

103

Then also each rectangle is a fiber product. Define (1, m) by the commutative diagram

ELD

A

Correspondingly, define (m,1) by

then by (2)

and

Thus ( fo ,fi) m( 1 , m) := ( fo ,fl) m(m,1). Since ( fo ,fl) is a monomorphism, the first diagram in Lemma 4 is commutative. For (ii) we use the commutative diagrams

2.

104

ADJOINT FUNCTORS AND LIMITS

A.

hence ( fo ,fl) m(efo , 1) = ( fo ,fl) implies again m(efo , 1) = 1A . Correspondingly, one shows m(1, efl) = 1, . For (iii) we use the commutative diagrams

and

Then fom(l,s) = f o P o ( 1 , s) = fo = foefo and flm(l, s) = flpl(l, s) = = fo = fief0 . Because of (fo,fl) m(1, s) = (fofl)efo , we also get m(1, s) = efo Again one shows m(s, 1) = efl correspondingly.

fis

.

Thus for a monomorphic equivalence relation there is a partially defined composition (on D _C A x A) on A which is associative (i), with neutral elements (ii), and invertible (iii). This is a generalization of Section 1.1, Example 16 to arbitrary categories. Compositions, that is, morphisms from a product A x A into an object A which have these and similar properties will be dealt with in more detail in Chapter 3. It is because of the properties proved in Lemma 4, that we use the name groupoid.

2.9

THE ADJOINT FUNCTOR THEOREM

105

2.9 The Adjoint Functor Theorem

PROPOSITION 1 . Let be a small category. Each functor 9E Funct(cc4, 9') is a colimit of the representable functors over S. Proof. We consider the following category: T h e objects are the representable functors over 9,that is, the pairs (hA, p') with a natural transformation p' : hA + F . T h e morphisms are commutative diagrams

where f : B + There is a forgetful functor (hA,p') I-+ *,(hf,p', 4) t-+ hf from this category into Funct(cc4, S ) , which we consider as a diagram. This diagram has a colimit by Section 2.7, Theorem 1, which is formed argumentwise and which is denoted by l @ h A . Furthermore, each q~ : hA --t 9 may be factored through 1Lm hA as hA + lim hA + 9. We show that the morphism T ( B ): 1% hA(B) + 9 ( B ) isTijective for each B E d. Let x E 9 ( B ) . Then by the Yoneda lemma there is an h" : h B + S with h"(1,) = x. Thus T(B)is surjective. Let u, v E 1% hA(B)with T(B)(u)= T(B)(v).Then there are C , D E d and y E hC(B)and x E hD(B)with y I-+ u under f : hC(B) 1% hA(B)and x I-+ v under g : hD(B)+ lim hA(B)by the construction of the colimit in S . Let cp : hC -+ 9and 4 : hD -+ 9be the corresponding morphisms into 9. Then v(B)(y)= $(B)(z).Thus by the Yoneda lemma, we get vhg = #h2 : h B -+ 9, that is, hB is over F with this morphism, and we get fh'(B)(lB) = u and gh2(B)(IB)= v. Hence, u = v and 7(B) is injective. If there are no natural transformations cp : h A + 9, then 9 ( A ) = 0 for all A E 'if. But we also have 1% hA(B)= o as a colimit over an empty diagram. T h u s we have also in this case S Ii+m hA. A

--f

COROLLARY I . Let sd be a small, finitely complete, artinian category. Let 9: d + S be a covariant functor which preserves finite limits. Then S is a direct limit of representable subfunctors. Prooj. We show that p;' : hA -+ 9may be factored through a representable subfunctor of 9. Let f : B + A be minimal in the set of subobjects of A such that there exists a commutative diagram

106

2.

ADJOINT FUNCTORS AND LIMITS

h A L 9

4/ hB

Then $ is a natural transformation. It is sufficient to show that

$(C): h B ( C ) + 9( C) is injective for all C E d .Let x, y E hB(C) be given with $(C)(x) = $(C)(y).Let D be a difference kernel of ( x , y ) . Since 2F preserves difference kernels, there is a commutative diagram hz h

C

Z hB-hD hY

*lJ 9

by the Yoneda lemma. Since D is a subobject of A up to equivalence of monomorphisms and because of the minimality of B, we get D B thus x = y. This implies that $ : hB + F is a subfunctor and that the element which corresponds to y : hA + F in g ( A ) is in the image of $(A). Consequently l@ h B = 9 if one admits for the h B only representable subfunctors of 9and if the colimit is directed. To prove that this colimit is directed let (hA,y ) and (he, $) be repreF ( A ) x 9 ( B ) ,we get sentable subfunctors of F. Since 9 ( A x B) Mor,(hAXB, 9) Mor,(hA, 9) x Mor,(hB, 9)

Thus there is exactly one p : hAXB 9, such that --f

is commutative. p may be factored through a representable subfunctor h C of 9. Let F : V + 9 be a functor. Let D E 9. A set S D of objects in V is called a solution set of D with respect to 9if to each C E V and to each morphism D + F C there is an object C' E !i?D and morphismsf : C' -+C and D --+ 9°C such ' that the diagram

D

-

9C'

2.9

THE ADJOINT FUNCTOR THEOREM

is commutative. If each D solution sets.

E9

107

has a solution set, then we say that S has

COROLLARY 2. Let V be a finitely complete category. Let F : V + S be a functor which preserves finite limits. Assume that there is a solution set for the one point set {a}= E with respect to 9. Then F is a colimit of representablefunctors. Proof. Let 2 = f?E be the solution set of E. Let 2 be the full subcategory of V with the set of objects 2.By Proposition 1, the restriction of F to 9is a colimit of representable functors on 9, that is li+m hA(B)= F ( B ) for A, B E 2. We want to prove that this equation holds for all B E V where the left side is argumentwise a colimit. First we reformulate the condition about the solution set. For each C E V and for each x E S(C),there is an A E 9 and an f : A + C and a y E F A with F f ( y ) = x, expressed differently: for each C E V and for each hx : hC -,9, there is an A E 9 and an f : A --t C and an hv : h* -,9 with hx ==hvht. This is a consequence of the Yoneda lemma. Since all the hA are over 9 and since lim hA(-) is a functor, we get a natural transformation T : 1% hA(-) + S through which the natural transformations hA + F may be factored. Furthermore, T ( B ) is an We want to prove this for all B E V. Let isomorphism for all B E 9. x E 9 B . Then there is an A' E 9,a morphism A' ---t B, and a y E S A ' which is mapped onto x by F A ' + S B . Since the diagram

-

lim hA(A')

lim hA(B)

1

1

--f

+

is commutative and since 1% hA(A') = F A ' , the morphism 1% hA(B)+ F B is an epimorphism. Let x, y E 1% hA(B)be such that they have the same image in F B . Then there are A', A" E Y with hA'(B) 3 u I+ x E ILm hA(B) and hA"(B)3 v tt y E ILm hA(B) and the images of u and v in F B coincide. Thus, hB

h" d

hA'

108

2.

ADJOINT FUNCTORS AND LIMITS

is commutative. Let C be a fiber product of u : A’ + B and v : A” --+ B. Then S C is a fiber product of S u and S v . Consequently, the diagram may be completed in two steps to the commutative diagram hB -hA’

where hA*--t S is the factorization of hC + S with A* E 3, which exists by the solution set condition. Thus the images of u and v are already equal in hA*(B) and consequently also in 1% hA(B). Hence, T(B)is an isomorphism. We observe that 9 C = o for all C E W if and only if the solution set for E is empty. In fact, the colimit over an empty diagram is an initial object. If V is empty, then the assertion of the corollary is empty, since then 9 is a colimit over an empty diagram of representable functors, that is, an initial object in Funct(%, S). COROLLARY 3 (Kan). Let A? be a small category, V a cocomplete category, and 9 : a2 + V a functor. Then there is a functor 3 : Funct(do, S) +V

which is uniquely determined up to an isomorphism such that d

Funct(do, S) 5V is commutative up to an isomorphism, that is, g h preserves colimits. g is left adjoint to the functor

9,and such that 9

-) : V + Funct(do, S)

Mor&F-,

with Morv(F-, - ) ( C ) ( A ) for the morphisms.

g

=

Mor,(9A, C) and an analogous formula

2.9

109

THE ADJOINT FUNCTOR THEOREM

Proof. By the required properties of 9 we get for a functor 3 E Funct(dO,S) with 3 = 1 5 h, (by Proposition 1) 9 ( X )= g ( 1 5 A,) E lim QhA

lim F A

+

---f

But 9 ( H ) = 1% 9 A defines a functor with the required properties, as is easy to check. Then Morw(9(X),C)

=

Mor%>(lim F A , C ) E lim Mor&FA, C) +

t

lim Mor,(h, , h c F ) t

=

g

Mort(#, Morc(9-,

= l i p h,F(A)

Mor,(lim hA , h#) +

-)(C))

shows the adjointness of 9 and Morv(9-, -).

THEOREM 1 (representable functor theorem). Let V be a complete nonempty category. A functor 9 : 5% S is representable if and only $ --f

(1) 9preserves limits ( 2 ) there is a solution set for { 0 } = E with respect to 9. Proof. Since 9 preserves empty limits, 9 preserves final objects. Thus there is a C E % with S C # o . By the preceeding corollary we know that 9is a colimit of the representable functors over 9where the representing objects are in the solution set 2. Let V : d -+ V be the functor which defines the diagram of the representing objects. Let B = lim V and u : X B -+ V be the f natural transformation of the projections. By the Yoneda lemma, a diagram hA

hf

hA'

is commutative if and only if 9f (#) = F. Let f : A' -+ A be a niorphism in the diagram defined by V . Let o(A) : B -+ A and u(A') : B -+ A' be the corresponding projection morphisms. Then we get two commutative diagrams hA

hf t

F

hA'

hA

hf hA'

hB

110

2.

ADJOINT FUNCTORS AND LIMITS

Since F is (argumentwise) a colimit of these representable functors, there is exactly one morphism q : 9+ hB with qhq = hu(A). We want to show that there is also exactly one natural transformation hP : hB-+ .F with hPhu(A) = hq. Since 9preserves limits, 9 Z 3 is a limit in the commutative diagram %=B

%=A'-

.rJ

%=A

For the elements y E 9 A and i,4 E F A ' used above, we get 9 f (#) = q ~ . Thus there is exactly one p E S B with .Fu(A)(p) = q ~ . Consequently, there is also exactly one hp : hB + 9 with h W A ) = h9. We only used that F preserves limits, which is also true for hB. T h u s h~ and q are inverse to each other and F is representable. Conversely, if 9 hB, then (2) is satisfied by B. (1) holds because of Theorem 2 of Section 2.7.

THEOREM 2 (adjoint functor theorem). Let V be a complete, nonempty category. Let 9 : V + 9 be a covariant functor. 9 has a left adjoint functor if and only if (1) S preserves limits, and (2) F has solution sets.

Proof. By Section 2.1, Proposition 2 S has a left adjoint functor if and only if Mor,(C, F-) is representable for all D E 9. But for a fixed D E 9 conditions (1) and (2) coincide with conditions (1) and (2) of Theorem 1 if we consider the reformulation of the solution set of E in Corollary 2. Thus, Theorem 1 implies this theorem. 2.10 Generators and Cogenerators For further applications of the adjoint functor theorem, we want to introduce special objects in the categories under consideration. A family {Gi}t,I of objects (with a set I) in a category V is called a set of generators if for each pair of different morphisms f,g : A + B in V there is a Gi and a morphism h : Gi-+ A with fh # gh. If the sets Mor,(G, , A) are nonempty for all i E I and all A E V then this definition is equivalent to the condition that the functor

n Morlg(Gi, -) i€I

: %+ S

2.10

GENERATORS AND COGENERATORS

111

is faithful. If the set of generators consists of exactly one element G, then G is called a generator. G is a generator if and only if Mor,(G, -) is faithful functor. If V is a cocomplete category with a set of generators {Gz}i,, and if all the sets Mor,(G, , A) are nonempty, then by Mor,(G,, -) E Mor,(U G i , -) the coproduct of the Gi is a generator.

n

LEMMA1. Let V be a category with a generator. Then the difference subobjects of each object form a set.

Proof. Let B and B’ be two proper difference subobjects of A. In the diagram

let (B, c) be a difference kernel of ( a , b). Let a’ = ad and b’ = bd. Now let d * Mor,(G, B’) = c Mor,(G, B ) as subsets of Mor,(G, A). For each f : G -+ B’, there is a g : G -+ B with cg = df; hence a’f = adf = acg = bcg = bdf = b‘f, This is true for each choice of f E Mor,(G, B’); hence a’ = b’. Consequently, there is exactly one h : B’ -+ B with ch = d . Analogously, one shows the unique existence of a morphism k : B -+ B‘ with dk = c. Thus c and d are equivalent monomorphisms defining the same difference subobject. Hence, the set of difference subobject has a smaller cardinality than the power set of Mor,(G, A), for different subobjects (B, c) and (B‘,d ) must lead to different sets d * Mor,(G, B’) # c * Mor,(G, B).

-

LEMMA2. Let V be a category with coproducts. An object G in V is a generator if and only if to each object A in V there is an epimorphism f:UG-+A.

Proof. Here we also admit a coproduct with an empty index set, which is an initial object. Let Mor,(G, A) = I. We form a coproduct of G with itself over the index set I. We define f : IJ G -+ A as the morphism with ith component i E Mory(G, A). Then f is an epimorphism if G is a generator. Conversely, if for each A there is an epimorphism f , then different morphisms g, h : A --f B stay different after the composition with f . But then for some injection G -+ G the composed morphisms must be different from each other.

112

2.

ADJOINT FUNCTORS AND LIMITS

LEMMA 3. Let W be a balanced category with finite intersections and a set of generators. Then W is locally small. Proof. As in Lemma 1 we shall show that different subobjects B and B’ of A define different subsets of Mor,(G, , A) for some i, where {G,} is a set of generators. Assume B and B’ different. Since V is balanced, not both morphisms B n B’ ---t B and B n B’ -+ B’ can be epimorphisms because in this case both would be isomorphisms compatible with the morphisms into A thus B = B’ as subobjects. Suppose B n B’ --f B is not an epimorphism. Then there exist two different morphisms f , g : B + C, such that (BnB’+BL+)

= (BnB’+B%c)

Let h : G, --t B be given with f h # gh. Then h cannot be factored through B n B’. Since B n B’ is a fiber product, there is also no morphism Gi ---f B’ with (Gg+B’+A)

=(GiLB-tA)

Thus the maps defined by B‘ + A and B + A map Mor,(Gi, B‘) and Mor,(Gi , B) onto different subsets of Mor,(G, , A) respectively. LEMMA4. Let d be a small category. Then Funct(&‘, S ) has a set of

generators. Proof. We show that {hAI A E d}is a set of generators. Let y, 4: 9-+ 93 be two different morphisms in Funct(d, S). Then there is at least one

A E d with v ( A ) # $(A). Thus by the Yoneda lemma, Mort(hA,9’) # Morf(hA,$), so there exists a p E Morl(hA,9) with yp # 4p. A cogenerator is defined dually. In S each nonempty set is a generator and each set with at least two elements is a generator. In Top each discrete, nonempty topological space is a generator and each topological space X with at least two elements and { 0 , X} as the set of open sets is a cogenerator. One also says that X has the coarsest topology. In S* each set with at least two elements is a generator and a cogenerator. In Top* each discrete topological space with at least two elements is a generator and each topological space with the coarsest topology and at least two elements is a cogenerator. We shall show more about the categories Ab, .Mod, Gr, and Ri in Chapters 3 and 4.

2.1 1 ADJOINT FUNCTOR

2.11

THEOREM

113

Special Cases of the Adjoint Functor Theorem

LEMMA.Let V be a complete, locally small category and let the functor 9: V ---t S preserve limits. For each element x E S C , there exists a minimal subobject C' C C with an element y E 9 C ' which is mapped onto x by the induced morphism 9C' -+ 9 C . Proof. Since 9preserves limits, the induced morphisms 9 C ' ---t .%=C are monomorphisms by Section 2.6, Corollary 5. T h u s the element y E 912is' uniquely determined. We consider the category of the subobjects B of C for which there exists a (uniquely determined) y E 9 B which is mapped onto x by 9 B -+ 9 C . T h e limit (intersection) C' of these subobjects has the same property because 9preserves limits, and because the existence of y E 9 C ' with this property is equivalent to the property that there exists a map { 0 } -+ 9 C ' which together with the map F C ' -.+ F C has the element x as an image. But this holds for the objects B in the above defined diagram.

THEOREM 1. Let V be a complete, locally small category with a cogenerator G.A functor 9: V -+ S is representable if and only if 9preserves limits. Proof. T o use Section 2.9, Theorem 1 we have to define a solution set for E. Let x E 9 C and let C coincide with the minimal subobject C' as constructed in the lemma. Let y E F G . If there is an f : C + G with F f ( x ) = y , then f is uniquely determined. In fact, if two morphisms have this property, then let D + C be the difference kernel of these two morphisms. Since 9 preserves difference kernels, there is an x' E 9 D which is mapped onto y by 9 D -+ P G . Since C is minimal (in the sense of the lemma), we get D = C, that is, both morphisms coincide. Thus we may consider Mor,(C, G) as a subset of 9 C . By the dual of Section 2.10, Lemma 2, C + G is a monomorphism, where the product is formed over the index set Mor,(C, G) and where the components of this morphism are all morphisms of Mor,(C, G). T h e n also C -+ C is a monomorphism where the product is formed over the index set F G and where we use for the additional factors of the product arbitrary morphisms of Mor,(C, G)as additional components. T h u s C is a subobject of n s r c G = D up to equivalence of monomorphisms. This holds for all such minimal objects C. Since V is locally small, these objects form a set, a solution set for E with respect to 9.

n

n

THEOREM 2. Let V be a complete, locally small category with a cogenerator. Let P : V' -+ 9 be a cocariant functor. 9has a left adjoint functor $ a n d only if S preserves limits.

114

2.

ADJOINT FUNCTORS AND LIMITS

Proof. This is shown in a way similar to the proof that Theorem 1 implies Theorem 2 in Section 2.9. COROLLARY. Let V be a complete, locally small category with a cogenerator. Then V is cocomplete.

Proof. Let d be a diagram scheme. The constant functor Z : V --f Funct(d, V) preserves limits. By Theorem 2, Z has a left adjoint functor lim. This holds for all diagram schemes d . + We now discuss an example for Theorem 2, where we refer the reader to textbooks on topology for the particular notions and theorems. The full subcategory of compact hausdod spaces in Top is a reflexive subcategory of the full subcategory of normal hausdorff spaces in Top. Urysohn's lemma guarantees that the interval [0, 13 is a cogenerator. The closed subspaces of compact hausdod spaces are again compact and represent the difference subobjects. By the theorem of Tychonoff, the products are also compact. Thus there is a left adjoint functor for the embedding functor. This left adjoint functor is called the Stone-Cech compactification.

THEOREM 3. Let V be a full rejexive subcategory of a cocomplete category 9. Then V is cocomplete. Proof. Let d :V + 9 be the embedding. Let d be a diagram scheme and 9 : d + V be a diagram. Let W : 9 + V be the reflector for 8. Since d is full and faithful, we get Mor,(9, 9')e Mor,(dG, 89') for 9,9' E Funct(d, V) which is natural in 9and 9' This . may be shown similarly to the isomorphism Mort(%#, '3) E Mar,(#, l@ 9) in Section 2.7, Theorem 1. Then the isomorphisms Morf(9, Z C )

Mor,(&9, 8°C)

Mor,(89, %&C)

N Morg(1im 89, 8 C ) E Moryp(W +

1 289,C)

are natural in 9and C. Thus V has colimits.

THEOREM 4. Let V be a full subcategory of a complete, locally small and locally cosmall category 9. Let V be closed with respect to products and subobjects in 9. Then V is a reflexive subcategory of 9. Proof. Since V is closed with respect to forming products and subobjects in 9,in particular with respect to difference subobjects, V is also

2.12

FULL AND FAITHFUL FUNCTORS

115

closed with respect to forming limits in 9 (of diagrams in V). Thus V is complete and the embedding functor preserves limits. Thus we have to find only a solution set. Since the embedding functor preserves limits, it preserves subobjects. Hence V is locally small. Given a morphism D + C. Since the functor Mor,(D, -) : V + S preserves limits, it preserves, by Lemma 1, a minimal subobject C‘ of C which may be factored through D + C. Let f , g : C‘ -+ D’ in 9 be given such that f h = gh, where h : D --t C’is the factorization morphism. Then h may be factored through the difference kernel of ( f,g ) . Since C’ was minimal, we get f = g and that h is an epimorphism. Consequently, the set of quotient objects of D is a solution set. Observe that we used in the proof only that V is closed with respect to forming difference subobjects instead of all subobjects. This, however, is often more difficult to check if one does not know exactly what the difference subobjects are. Some examples are that the full subcategory of commutative rings is a reflexive subcategory of Ri. Similarly, the full subcategory of hausdofi spaces in Top is reflexive. We also observe that the full subcategory of integral domains is not reflexive in Ri, for if it were it would have to be closed with respect to forming products in Ri. But the product of Z with itself is not an integral domain.

2.12 Full and Faithful Functors LEMMA 1. Let 9: ‘3--+ 9 be a faithful functor. Then 9 reflects monomorphisms and epimorphisms. Proof. Given f , g , h E $? with f g = f h . Then 9 f 9 g = 9 f 9 h . If 9 f is a monomorphism, then 9 g = 9 h . Since .Fis faithful we get g = h. By dualizing, we get the assertion for epimorphisms.

PROPOSITION 1. Let 9 : V + 9 be a full and faithful functor. Then 9 reflects limits and colimits. Proof. Let 9 : d -+ V be given. 9 has a limit if and only if the functor Morj(X-, 9): go --+ S is representable. Given C E V with Morj(X-, 99) g Mor,(-, S C ) . Then M o r , ( 9 Z - , S g ) g Mor,(S-, 9 C ) as functors from V ointo S. Since S is full and faithful, we get Mor,(,X-, 9) Mor,(-, C). Dually, one shows that 9 reflects colimits.

116

2.

ADJOINT FUNCTORS AND LIMITS

PROPOSITION 2. Let d be a small category. The covariant representation functor h :a? -+ Funct(dO,S) reflects limits and colimits and preserves limits. Proof. We know from Section 1.15 that h is full and faithful. Thus Proposition 1 holds. The last assertion is implied by Section 2.7, Theorem 2. Observe that h does not necessarily preserve colimits. In fact, let &’ be a skeleton of the full subcategory of the finitely generated abelian groups in Ab. Then d is small. We may assume that H and Z/nZ are in a? for some n > 1. Then Z/nZ is a cokernel of n : H -P Z, the multiplication with n. But Mor,(--, H/nH) is not a cokernel of Mor,(--, n) : Mar,( -, H) -+ Mor,( -,H) because this does not hold argumentwise, for example for the argument H/nH.

PROPOSITION 3. Let 9 :V -+ 9 be left adjoint to 9 : 9 V . Let y5 : Mor,(-, 9-) E Mor,(9-, -) be the corresponding natural isomorphism and let Y : 9 9 -+ Id, be the natural transformation constructed in Section 2.1. Then the following are equivalent: --f

(1) S is faithful. (2) S reflects epimorphisms.

( 3 ) If g : C --+ S D is an epimorphism, then also $(g) is an epimorphism. (4) YD : 9 S D -+ D is an epimorphismfor all D E 9. Proof. That (1) a (2) is implied by the lemma. (2) 3 (3): By the remark after Section 2.2, Theorem 1, S g * = 9 ( $ ( g ) )is an epimorphism if g is an epimorphism. Then by (2), $(g) is also an epimorphism. (3) a (4) holds if one sets for g the identity l g D .(4) * (1): The map ’3 : Mor,(D, D’) -P Mor,(SD, GD’) is by definition of Y : 9 9 -+ Id, composed by Mor,(D, D’) -+ Mor,(9SD, D‘) g Mor,(SD, 9D‘). If YD is an epimorphism, then this map is injective. LEMMA 2. With the notations of Proposition 3 , 9 is full if and only i f the morphisms YD :9 9 D -+ D are sections. Proof. We use Section 1.10, Lemma 3 and the fact that the map 9 : Mor,(D, D’) -+ Mor,(BD, 9D’) is composed of Mor9(D, D’) + Mor&FYD, D’)

MorW(SD,YD’)

2.12

FULL AND FAITHFUL FUNCTORS

117

COROLLARY.With the notations of Proposition 3, 9 is full and faithful and only if the morphisms YD :9 9 D -+ D are isomorphisms.

Proof. This corollary is implied by Proposition 3 and Lemma 2 because the isomorphism between Mor,(D, D’) and M o r B ( 9 9 D , D’) for all D’ (natural) implies the isomorphism between D and 9 9 D . PROPOSITION 4. With the notations of Proposition 3, let 9 be full and faithful. Let X ; d -+ 9 be a diagram. Let % be a limit or a colimit of 9s. Then F C is a limit or, respectively, a colimit of X . If V is(Jinitely) complete or cocomplete, then 9 is also (jinitely) complete or cocomplete respectively.

Proof. Since in the case of the colimit, Mor,(C, -) we get Mor9(9C, -)

Morop(C, 8-)

Morf(8X, 9X-)

Mor,(9X, X - ) , Mor,(X, X - )

We prove the second assertion in the inversely connected category Vx’( F u n c t ( d , V), V), where we get a commutative diagram

T h e morphism (9YX)(@9&)is the identity. Since C is a limit, there is a uniquely determined morphism p, and p(@C) is also the identity. T h u s p is a retraction. Since @ : Id, -+ 9 9 is a natural transformation, and since 99@ = @99by (9Y9)(9F@) = (9YF)(@99), the square 9 F C P - c

1

99-(@C)

8989C

-..1 ??SP

8FC

is commutative. Since ( 9 F p ) ( 9 9 ( @ C ) is ) the identity, p is an isomorphism, hence also @C.Since 9YX is an isomorphism, also @9Z is an isomorphism. Thus B F C is a limit of 999Z.9,being full and faithful, This proves the second reflects limits. Thus S C is a limit of 99s. assertion of the proposition.

2.

118

ADJOINT FUNCTORS AND LIMITS

Problems 2.1. Let Y : Gr --t S be the forgetful functor which assigns to each group the underlying set. Formulate the universal problem in V 9 ( S , Cr) for A E S and determine whether a universal solution exists. Does Y have a left adjoint functor ? Formulate the universal problem in Vs(Gr, S). How does the universal solution change if one replaces Gr by Ab ?

2.2.

If

AI-B

4 La B A C is a fiber product, then f is a monomorphism.

2.3.

A full faithful functor 9 defines an equivalence with the image of 9.

2.4,

If P : V

-+

S has a left adjoint functor, then 9 is representable.

2.5. Prove (without using Section 2.8, Lemma 3) that each kernel pair is a monomorphic equivalence relation. 2.6. (Ehrbar) Let 9 and Y be subcategories of a category Q. We say that 2 and 9’ decompose the category V if all objects and all isomorphisms of Q are in 1as well as in 9, if there is a 9-9’-decomposition for each f~ Q, that is, if to each f~ V there is a pair (q, s) E 9 x 5” with f = sq, and if to any two 1-9’-decompositions (q, s) and (q’, 5’) of the same morphismfe Q there is exactly one h E Q with hq = q’ and s = s’h. Show that h is an isomorphism. then there is exactly one morphism d E Q such that If bq = sa with q E 1 and s E 9,

the diagram

C

A

D

is commutative. Let f~ V and A E V. f is called an epimorphic relative to A if Morr(f, A ) is an injective map. Let Q be a category with nonempty products and assume that Q is decomposed by the subcategories 9 and 9’.Let ‘ube a class of objects in V with the property that all q E 9 are epimorphic relative to all A E ‘u.Let %* be the full subcategory of Q with the C A and a morphism s : A* --* Hi,, A , objects A* E V for which there is a family with s E 9’.‘u*is a reflexive subcategory of Q if and only if to each object B E Q there exists a nonempty set L of morphisms f E Q with B the domain off and with the range off in ‘u*and with the property that to each g E Q with B the domain of g and the range of g in ‘uthere is an f s L and an h E Q with g = hf. (Hint: Since Y contains products of morphisms, X* contains products. Furthermore, all q E 9 are epimorphic relative to all A * E U*. I f L is as above, and if h : B --t &L R(f)(with R ( f )the range o f f ) is the morphism with p,h = f for all f EL, and if (q, s) is a 1-9’-decomposition off, then q is

119

PROBLEMS

the adjunction @(B) with Q : Idy + 1where W is the reflector we wanted to find (Section 2.1, Theorem 1 and 2 and Section 2.4).) Q = Top, 2 the category of continuous, dense maps in Q, and Y . the category of injective, closed, continuous maps define the Stone-Cech cornpactification with = {[O, 111. 2.7.

(a)

Use the construction in the proof of Section 2.6, Proposition 2 to show that for a diagram .F : d

(b) that for a diagram 9 : d lim F

-----,

=

I-

-+

S the limit of .F is

+

S the colimit of F is

equivalence classes in xA

u

.FA (disjoint union) with

A€&

( ~ j ) ( x g ) for

all

f :A

+

B in S ; x A E .FA;

xB E

.FBI

(c) that for a directed diagram scheme d and a diagram .F : .d 4 S with Ffsurjective for all f E d the direct limit is

3 Universal Algebra The theory of equationally defined algebras is one of the nicest applications of the theory of categories and functors. Many of the wellknown universal constructions, for example, group-ring, symmetric and exterior algebras, and their properties can be treated simultaneously. The introduction into this theory in the first two sections originates from the dissertation of Lawvere. T h e method of Section 3.3 leads to Linton's notion of a varietal category, which, however, will not be explicitely formulated. In the fourth section we shall use the techniques of monads or-as they are called in Zurich-triples. Theorem 4 in the last section is essentially a result of Hilton.

3.1

Algebraic Theories

Let N be the full subcategory of S with finite sets as objects, where for each finite cardinal there is exactly one set of this cardinality in N. In particular, let 0 be in N. We denote the objects of N by small Latin letters (n E N). In special cases we shall also use the cardinals of the corresponding sets as objects of N (0, 1, 2, 3,... E N). Let n EN.Then n is an n-fold coproduct (disjoint union) of 1 with itself. 0 (= 0 ) is an initial object in N (empty union). Consequently, we get Mor,(m, n) Mor,(l, n)m (m-fold product). Since each morphism 1 -+ n is an injection into the coproduct n, all morphisms in N are m-tuples of injections into coproducts. N is a category with finite coproducts. Let No be the dual category of N. T h e objects will be denoted just as in N.Each object n E Nois an n-fold product of 1 with itself. 0 is a final object. Each morphism is an n-tuple of projections from products. In particular, we identify MorNO(m,n) = MorNO(m, 1)" (n-fold product). Nois a category with finite products. A covariant functor A: No -+ 9l which is bijective on the object classes 120

3.1

ALGEBRAIC THEORIES

121

and which preserves finite products is called an algebraic theory. I n particular, A preserves the final object. Since A is bijective on the objects, we denote the corresponding objects in % and in No with the same signs, that is, with small Latin letters or the corresponding cardinals. pni : n -+ 1 denotes the ith projection from n to 1 in No as well as in a. We shall often talk about an algebraic theory % without explicitly giving the corresponding functor A since this functor may be easily found from the notation used. Let A : NO + % and B : No -+ !I3 be algebraic theories. A morphism of algebraic theories is a functor 9 : --+ 23 such that 9 A = B holds. Thus the algebraic theories form a category Alt. An algebraic theory A : No -+ % is called consistent if A is faithful. Let Jlr be a discrete category with a countable set of objects denoted by 0, 1, 2, 3,... . Let % : Alt -+ Funct(Jlr, S) be a functor defined by %(A, %)(n)

=

Mor&

1)

B(g)(n) = ('3 : Mor%(n,1) -+ MorB(n, 1)) % has a left adjoint functor 5 : Funct(Jlr, S) -+ Alt. THEOREM.

Proof. We construct 8 explicitly. Given H : Jlr -+ S. We construct sets M ( r , s) for r , s E Nin the following way. First let M&, 0 ) = (4 M1(r, 1)

=

M1(y,s) =

H ( r ) u MorNo(r,1) M1(r,l)s

for s

with a disjoint union

>1

We denote the s-tuples also by (ul,..., u,). Then define

In contrast to the s-tuples in Ml(r, s) we write the pairs [a,T] with brackets. If the sets M i - l ( ~s), and Mi-l(r, s) are already known, then let

3.

122

UNIVERSAL ALGEBRA

Then

c Ml(Y,0)c Ml’(Y, 0)c

{my}

0)c Mz’(Y, 0)c *..

Mz(Y,

H ( Yu ) MorNo(r,1) C M1(y, 1) C M1’(r,1) C M&, 1) C M;(Y, 1) C

Ml(Y,s) c Ml’(Y, s)

c M&, s) c Mi(Y,s) c .*.

hold. So we define M(r, s) = U Mi(r, s). T h e following assertions hold: (a) {wr} C M(r, 0) for all I 2 0. (b) H(r) u MorNO(r,1) C M(Y,1). (c) If uiE M(r, 1) for i = 1,..., s with s > 1, then the s-tuple (ul ,..., u,) E M(r, s) for all r 2 0. (d) If u E M ( t , s) and t

T

M(Y,t ) , then [u,T ] E M(r, s) for all

E

Y,

s,

0.

On the sets M(Y,s) let R be the equivalence relation induced by the following conditions: (1) If u, T E M(r, 0), then

(2) If ui E M(r, 1) for j i = 1,..., s. (3) If 0 E M(r, $1, then

(0, T )

=

E R.

1,...,s, then ([p,i, (al ,...,OJ], ui)E R for

(([P,’,GI,..., [P,”,41, 0) E R.

(4) If u E M(r, s), then

([(P:,..., Ps”, 4, 0)E R ( 5 ) If u E M(r, s),

7

E

and

([O,

(P?,...,P r r K 4 E R.

M(s, t), and p E M ( t , u), then

(“PI 7l,ul, [P,

[T,

41) E R.

( 6 ) If ui ,T~ E M(r, 1) and (ui , T ~E) R for i ((01

,..*, us), (71,. *,

(7) If u, u’ E M(r, t) and ([T,01, b’, 0’1) E R.

T , T’

7s))

=

1,..., s, then

E R.

E M ( t , s) and (u, u‘),

(7, 7 ’ )

E

R, then

Observe that two elements are equivalent only if they are in the same set M ( Y ,s). Thus we define Morgn(r, s) = M(Y,s)/R as the set of

3. I

ALGEBRAIC THEORIES

123

equivalence classes defined by R. Let $ E MorgH(r, s) and rp E MorgH(s, t ) with the representatives T E Mi(r, s) and u E Mi($, t ) which is possible by a sufficiently large choice of i. Then let the composition q ~ $ of q~ with 4 be the equivalence class of [u,T] E Mi+l(r,t ) C M(r, t ) . By (7), this class is independent of the choice of the representatives of rp and $. By (3, this composition is associative. By (4), the equivalence class of (p:, ...,pVr) is the identity for the composition. T h u s g H is a category with the objects 0, 1, 2, ..., and the morphism sets MorSH(r, s). 0 is a final object in EH by (1). Conditions (2), (3), and (6) imply Morg,(r, In fact, (6) implies that for morphisms MorgH(r, s) rpi E MorgH(r, I ) with representatives uiE M(r, 1) for i = 1,..., s the morphism (yl ,..., rps) E MorgH(r, s) with the representative (ul ,..., us)E M(Y,s) is independent of the choice of the representatives ui . (2) implies the existence of a factorization morphism rp such that pniv = rpi , namely rp = (rpl ,..., rps) and (3) implies the uniqueness of such a factorization morphism. Thus the object S E 3 H is an s-fold product of 1 with itself. Obviously s I+ s and pii I+p j i induces a product-preserving functor No-+ g H , called the free algebraic theory generated by H. Let H, H' E Funct(M, S) and let f : H -+ H' be a natural transformation. Since M is discrete, the mapsf(r) : H(r) -+ H'(r) may be chosen arbitrarily. Define gf by

Then gfmaps the equivalence relation R into R'. Hence, gfis a morphism of free algebraic theories gf : g H -+ gH'. One easily verifies g(fg) = g(f )g ( g ) and 51, = Id,, . Thus 5 : F u n c t ( N , S) -+ Alt is a functor. It remains to show that

holds naturally in H and (A, a).Let f : H -+ %(A, 9l) be given, that is, for each Y letf(r) : H ( r ) -+ Mor,(r, I ) be given. We define a morphism g : g H -+ (A, a)of algebraic theories in the following way.

124

3.

UNIVERSAL ALGEBRA

First let g(w,.) E Mora(r, 0) g(4

=

f (4

for

w,. E M(Y,0)

for all

dPi") = Pr" g ( ( q ,..., aJ) = x E Mora(r, s)

a E H(Y)

for all and all

where x corresponds to the Mor,(r, s) E Mor,(r, 1)5, and g ( b , 71)

ai E M(r, l), Y

i = 1 ,..., s,

EM (g(u,), ..., g(ag)) under

element

=g ( 4g ( 4

for all (T E M(t, s), 7 E M(r, t) and all Y , s, t E M . Thus g : M(Y,s) -+ Mor,(r, s) is defined. Since a is an algebraic theory, R-equivalent elements in M(Y,s) are mapped into the same morphisms in Mor,(r, s). Thus we get a functor g : 5 H -+ (A, a)which is a morphism of algebraic theories because of g( p:) = p,i. If, conversely, g : 5 H + (A, a)is given, then we get a family of maps f ( ~: )H(Y)-+ MorgH(r, 1) -+ Mor,(r, 1). These two applications are inverse to each other and compatible with the composition with morphisms H -+ H' and (A, a) -+ (B, d),hence natural in H and (A, a). Let two morphisms of free algebraic theories p , , p , : 8~5 -+ 5 H be given. If one extends the equivalence relation R which we used for the construction of S H by the condition

(8) If v E MorgLh

4,then ( P l ( V ) , Pz(v)) E R

then the equivalence classes for this new equivalence relation form again an algebraic theory. This may be seen in the same way as in the construction of free algebraic theories. Conversely, let '3 be an algebraic theory and Y : SB(2I) -+ a be the adjunction morphism of Section 2.1, Theorem 1.

*

U n ) = {(P,4) I P, and

E MO'SBD(a)(%11,

VP) =

yw))

3.1

ALGEBRAIC THEORIES

125

define morphisms qi :L -+ BGB(%).Since 5 is left adjoint to 23, we get morphisms pi : SL --f BB(2l). Since B(Y) q, = B(Y) q2 holds for

0(a)

L 22@B(%)

the equation Yp, = Up, holds for iyL 2,@I(%) % the functor Y is surjective on the Because Mor,(r, s) E Mor,(r, morphism sets. If Y(p) = Y(#),then y , 4 E h%p3(%)(r, s) e Morgcn(cu)(r,

and hence ptY(9) = p:Y(#). But then p t p , pB# E Morgm(%)(r, 1) with !P(p,"v)= Y(p>#).Consequently, p,ip and p i # are equivalent for

i = 1,..., s with respect to the equivalence relation extended by (8). Also p and # are equivalent by (2) and (6). T h u s this new equivalence relation defines an algebraic theory isomorphic to 2l. Thus each algebraic theory may be represented by giving H , L E F u n c t ( N , S), and two morphisms q, , q2 : L --t BGH (instead of p, , p , : GL + 5 H ) . One may choose L(n) as above as pairs of elements in BBH(n), such that ql(n) and q,(n) may be defined as projections onto the particular components. In the following we shall always proceed in this way. T h e elements of H ( n ) are called n-ary operations, the elements of L(n) identities of nth order. Obviously one can use different n-ary operations and identities of nth order for the representation of the same algebraic theory. Thus also the elements of B%(n) are called n-ary operations. Example An important example is the following representation. The represented algebraic theory is called the algebraic theory of groups. 71

H(n)

H(n) = L(n) =

L(n)

0

for n 2 4

126

3.

UNIVERSAL ALGEBRA

Explicitly this scheme for the algebraic theory of groups means that there exist morphisms e : 0 -+ 1, s : 1 -+ 1, and m : 2 -+ 1 such that the following diagrams are commutative: (1

,I)

1 L l x l

Im

O11

0

2

1

\ I1

mX 1

1x1X l A 1 x 1

1

hXm

1 x 11-

m

lm

where 0, : 1 - 0 is the morphism from 1 into the final object 0 and where 1, x m = (p31, m(p32,P ~ ~and ) ) m x 1, = (m(p3', P ~ ~ ~)3,3 ) . If one interprets e as the neutral element, s as forming inverses, and m as multiplication, then the diagrams represent the group axioms.

3.2 Algebraic Categories Let % be an algebraic theory. A product-preserving functor A : ' 3 3 S is called an %-algebra. A natural transformationf : A + B between two '%-algebras A and B is called an '%-algebra homomorphism or simply an %-homomorphism. The full subcategory of Funct('%, S) of productpreserving functors is denoted by Funct,(%, s) and is called the algebraic category for the algebraic theory '%, An %-algebra A is called canonical if A(n) = A( 1) x *.* x A( l), where the right product is the set of n-tuples with elements of A(l), and if A(pni)(xl,...,xn) = xt for all n and i . Let the algebraic theory 9l be represented by H and L, and let A be a canonical %-algebra. Then A induces a product-preserving functor B : BH + '% + S which is a canonical BH-algebra. Let rp be an n-ary operation of H(n), and let A(l) = B(1) = X.Then the map B ( f p ) : X x -*.

xx-x

3.2

127

ALGEBRAIC CATEGORIES

is an n-ary operation on the set X in the sense of algebra. Let ('p, $) EL(n) be an identity of nth order. Then the two operations B('p) and B($) coincide on the set X, though the n-ary operations 'p and $I in BH may be different. Thus an identity (or equation) for the operations on the set X is given. T h e %-algebra A is called an equationally defned algebra. If % is the algebraic theory of groups and A a canonical %-algebra, then A is a group. T h e maps A(e) : { 0 ) + A(I), A(s) : A(1)

--+

A(1),

and A(m): A(1) x A(1) --+ A(1)

interpreted as neutral element, inverse map, and multiplication respectively make the following diagrams commutative

A(1) x A(1) x A(l)-A(l)

x 1,m)

x A(l)

since A is a functor. Hence A( 1) is a group. Conversely, if G is a group with the multiplication p : G x G -+ G, the neutral element E : {a}+ G, and the inverse a: G-+ G, then we define A(n) = G x x G (n times), A(m) = p, A(e) = E , and A(s) = u. If we represent the algebraic theory % of groups as in Section 3.1, then these data suffice to define uniquely a canonical SH-algebra A: SH -+ S. Since G is a group, all the identities of L hold for this BH-algebra. So this defines, in fact, a canonical %-algebra. This implies the following lemma.

LEMMA1. There is a bijection between the class of all groups and the class of all canonical %-algebras, where % is the algebraic theory of groups.

3.

128

UNIVERSAL ALGEBRA

Let f : A + B be an %-homomorphism of canonical %-algebras. Let q~ : n + 1 be an n-ary operation in %. Then the following diagram is

commutative: A(1) x

fU)X

***

x A(l)-B(l)

...Xf(l)

x

... x B(1) -1.w

b(p.)

f(1)

'B(1) In fact, one easily verifies with the operations pnl,...,pnn that f ( n ) = 41)

f(1) x x f(1). Iff is a map from A(l) to B(l) such that the above diagram is commutative for all n and all n-ary operations q ~ ,then f is an %-homomorphism. Thus the %-homomorphisms are homomorphisms in the sense of algebra, compatible with the operations. So it suffices to give a map f : A( 1) + B( 1) compatible with the n-ary operations in H ( n ) for all n, if one defines f ( n ) = f x x f. Then f is already an %-homomorphism. This follows directly from the definition of S H . For the example of the algebraic theory of groups, this means that the group homomorphisms may be bijectively mapped onto the %-homomorphisms of the corresponding %-algebras and, consequently, that the category of groups is isomorphic to the full subcategory of the canonical %-algebras of Funct,(%, S).

LEMMA 2. Let % be an algebraic theory. Then each %-algebra A is isomorphic to a canonical %-algebra B in Funct,,(%, S ) . Proof. Let B(l):= A(1) and B(n) := B(1) x x B(1). Let B(pni) be the projection onto the ith component of the n-tuples in B(1) x *.. x B(1). Then B(n) is an n-fold product of B(1) with itself. Thus there exist uniquely determined isomorphisms A(n) B(n), such that for all projections the diagram A(n)s B(n) A(P,91

b(P"')

A(1) = B(1)

is commutative. Let cp : n + 1 be an arbitrary n-ary operation in %. Then B ( ~ Jis)uniquely determined by the commutativity of

4)= B(n) A w l

A(1)

b(v) = B(1)

3.2

ALGEBRAIC CATEGORIES

129

I t is easy to verify that B is a canonical %-algebra, which then by construction is isomorphic to A. Using Section 2. I, Proposition 3 we obtain the following corollary.

COROLLARY 1. Let A be the algebraic theory of groups. Then Funct,(A, S ) is equivalent to the category of groups. The full subcategory of canonical %-algebras is isomorphic to the category of groups. T h u s far we have discussed only the example of groups in detail. But similar considerations hold for each category of equationally defined algebras in the sense of (universal) algebra, in particular the categories S, S*, Ab, ,Mod, and Ri. For Ri choose for a representation of the corresponding algebraic theory the 0-ary operations: 0, I I-ary operation:

-

2-ary operations: f,

-

+,

T h e identities are, apart from the group properties with respect to the associativity and the distributivity of the multiplication, the commutativity of the addition, and the property of 1 as the neutral element of the multiplication. T h e reader can construct the corresponding diagrams easily. S is defined by H = 0 and L = 0 . Thus the corresponding algebraic theory is NO. Another interesting example is ,Mod. Here the operations are e, s, and m for the group property and, in addition, all elements of R considered as unary operations. Hence this is an example where H(l) may be infinite. T h e identities arise as in the above example for rings from the defining equations for R-modules. Let Funct,,(BI, S) be an algebraic category. T h e evaluation on I E % defines a functor B : Funct,(%, S) -+ S with B(A) = A(1) and B( f ) = f (1). This functor will be called the forgetful functor. T h e set B(A) = A( 1) is called the underlying set of the %-algebra A.

THEOREM. Let % be an algebraic theory. The algebraic category Funct,,(%, S ) is complete, the limits are formed argumentwise, and the forgetful functor into the category of sets preserves limits and is faithful. Proof. By Section 2.7, Theorem 1 Funct(%, S) is complete and the limits are formed argumentwise. Since limits commute with products, a limit of product-preserving functors is again product preserving. Since

130

3.

UNIVERSAL ALGEBRA

the forgetful functor is the evaluation on 1 E A and since limits are formed argumentwise, 23 preserves limits. Let f, g : A -+ B be two %-homomorphisms and let f(1) = g(l), then f (n) = g(n) for all n E N, since all diagrams

a B(n)

A(n)

bbn')

Abn')!

A(l)%B(l)

are commutative. Consequently, 23 is faithful. COROLLARY 2. Let f : A + B be an %-homomorphism of %-algebras. f is a monomorphism in Funct,(%, S ) if and only iff (1) is injective.

Proof. 23, being faithful, reflects monomorphisms (Section 2.12, Lemma 1). 23, preserving limits, preserves monomorphisms (Section 2.6, Corollary 5).

A subobject f : A + B is called a subalgebra. The corollary implies that Funct,(%, S)is locally small since 23 is faithful and S is locally small. The Theorem and Corollary 2 are generalizations of some assertions we made in Chapter 1 for S, S*,Gr, Ab, Ri, and .Mod. The example Z --f P in Ri of Section 1.5 shows that epimorphisms in Funct,(%, S) are not necessarily surjective maps (after the application of the forgetful functor). So the example in Section 1.5, which shows that in Gr (and also in Ab) the epimorphisms are exactly the surjective maps, becomes all the more interesting. 3.3 Free Algebras Let A : NO -+% be an algebraic theory. We construct a productpreserving functor A, : SO + %, which is bijective on the object classes, and a full faithful functor & : % + N, such that the diagram A

No-

N

is commutative where No+ So is the natural embedding. We may identify the objects of %, with the objects in So. For two sets X and Y,

3.3

131

FREE ALGEBRAS

we define Moram(X, Y) = Morgm(X, 1 ) Y . Then A, will become a product-preserving functor. For the definition of Moram(X, 1) let X * be the set of triples (f,n , g ) wheref : X -+ n is a morphism in So and where g : n -+ 1 is a morphism in 8 . Here n is a finite set in NO. We call two elements (f,n , g ) and (f‘,n’, g‘) in X* equivalent if there is a finite set n” in Noand if there are morphisms X + n”, n” -+ n‘, and n’’ -+n in So such that the diagrams n

ift A1

X-

n”

in So

n‘

and

n‘

are commutative. This relation is an equivalence relation. We only have to show the transitivity. Let (f,n, g ) (j’, n’, g ’ ) and (j’, n‘, g’) (f”,n”, g”) and let n* and n** be elements which induce the equivalences. Let m be the fiber product of n* -+ n’ with n** -+ n’. Then the diagram

-

-

132

3.

UNIVERSAL ALGEBRA

is commutative. (Compare the morphisms in the corresponding categories.) Let Morxm(X, 1) be the set of equivalence classes.

LEMMA 1. ‘illm is a category.

Proof. Let (f,n,g ) be a representative of an element in Morwm(X,1). ThenfO is a map from n into X in the category S. Let n’ be the image of n under this map. Then we may decompose f : X + n as follows f‘ X --f n’ -P n. Obviously then (f’, n’,gh) is equivalent to (f,n,g). Furthermore, n’ is (up to equivalence of monomorphisms) a finite subset of X. Such a representative will be called reduced. n’,g‘) : Y -+ I be reduced Let ((f,,ni ,g,),,,) : X + Y and (f’, . Let Y = n, (disjoint union or representatives of morphisms in ‘illm coproduct in N). Then by the product property of I in So, the following morphisms are defined: f : X - t I by the f i and g : r + n‘ by the g, Then let the composition of the given morphisms be (f,r, g’g). This composition still depends on the choice of the representatives. Let ( f”,n”,g”) be reduced and equivalent to (f’, n’,g‘). Without loss of generality we may assume that n’ C n” C Y in S and that hf” = f‘ and g‘h = g” for h : n” -+ n‘ in So. Let r’ = n, , then

.

xien* n”

Y’

Y

-n’

is commutative. Similarly one shows that the composition does not depend on the choice of the representatives of the (f, , ni ,g,). Let p, : X + 1 be the projections from X into 1 in So. Then ( ( p , , 1, is the identity on X in ‘illm . In fact, given (f,n,g ) : X + 1, then (f,12, g)((Pz 1, 11)ZEX) = (f,12, g). Given ((ti n, gz)r.x) : I,+ then (P, > 1 9 11)((h ? n, 7 g i ) i E X ) = (f, n, g,). T o prove the associativity let ((fy , ny ,g&,) : X -+ Y, 9

9

9

((fs

9

nz

,gz)xsz) : y

-+

z,

x,

9

9

and

(f,n,g) : z

-+

1

be reduced representatives of morphisms in ‘illm. It is easy to see that

c c n,, c ny =

with

implies that the composition is associative.

Y

=

n,

3.3

133

FREE ALGEBRAS

COROLLARY. There exists a product-preserving functor A, : So+ 21m which is bijective on the classes of objects and a full faithful functor 9,: 2l-+ 21m such that

is commutative. Proof. I t suffices to define A, on the projections p , : X - t 1. Let A,(p,) = ( p , , 1 , 1J. Then it is clear that A, preserves products. Let 9,(n) = n and 9,(g) == (1, , n, g ) for g : n + 1 in 2l. Since we have (1, , n, g ) ( g , 1, 11) for g : n -+ 1 in NO,the square is commutative. We still have to show that 9,is full and faithful. Given f , g : n + 1 (1, , n, g ) in 21m . Then there exist n’ and in 2I. Let (I,, n,f ) 1 : n + n’ with a commutative diagram N

N

1

Hence hl = 1, and k l = 1,. Furthermore, f h = gk. B y composition with 1 we then get f = g. Thus 9 ,is faithful. Now let n‘ n -% 1 in 21a be given. Then (f,n, g ) (I,, , n’, g f ) and &,(gf) = (1,t , n’,gf). Hence 9 ,is full and faithful.

-

LEMMA 2. Let A : 2l+ S be a product-preserving functor. Then there exists up to an isomorphism exactly one product-preserving functor A’ : a, + S with A’& = A. Proof. In order that A‘& = A and that A’ preserves products, A’(l)X and A’(1) = A(1). Furthermore, we must have A’(X) A‘(p, , 1, 11) A ( 1 ) P z and A‘(ln, n, g ) = A ( g ) must hold. B y composition A’(f , n, g ) E A ( g ) A ( l ) f must hold. With these definitions, A’ is a product-preserving functor and, in fact, A’$, = A holds.

134

3.

UNIVERSAL ALGEBRA

LEMMA 3. Let A, B : 81 + S be product-preserving functors and A', B' be the extensions to %a as constructed in Lemma 2. Let cp : A -+ B be a natural transformation. Then there is exactly one natural transformation cp' : A' + B' with ~ '= 9 9). ~ Proof, We define v ' ( X ) g ~ ( 1 : )A(l)X ~ -+ B(I)*. Obviously this is the only possibility for a definition of v' because the functors A' and B' preserve products. At the same time it is clear that cp' behaves naturally with respect to all projections between the products. But cp' is natural also with respect to the morphisms in 2l,since we only have to consider the restriction y'YW= 9.

THEOREM 1. Let 9l be an algebraic theory. The forgetful functor V : Funct,(%, S) + S is monadic. Proof. We define a functor 9: S --t Funct,(%, S) by g(X)(-) Morflm(X,-). Then Mors(X, V A ) N

Mors(X, A( 1)) g A( l)X Mor,(MorAm(X,-), A')

=

A'(X) = Morr(S(X), A )

holds naturally for X E S and A E Funct,,(%, S)where we used the last two lemmas. Now we use Section 2.3, Theorem 2. Let f o , f i : A + B be a V contractible pair in Funct,(%, S). Since there are difference cokernels in S we get a commutative diagram in S:

C(1)

ICCll

'C(1)

where we wrote f t instead of &(1). If we form the n-fold product of all

3.3 FREE ALGEBRAS

135

objects and morphisms of this diagram, we get again a corresponding diagram. In particular

is a difference cokernel. Given p : n -+1 we get a commutative diagram

where C(p) is uniquely determined by the property of the difference cokernel. Thus C : B --t S with C ( n ) := C(1)" is a product-preserving functor and h : B -+ C a natural transformation which is uniquely determined by h(1) : B(1) -+ C(1). Since C(n) is a difference cokernel for all n E B, C is a difference cokernel of (fo,f i ) in Funct,(%, S ) . This theorem shows that the %-algebras and %-homomorphisms are exactly the Y9-algebras and YF-homomorphisms in the sense of Section 2.3. Thus the free Y9-algebras are also called free %-algebras. 9 ( X ) is called free %-algebra freely generated by the set X .

PROPOSITION. Let 2f be the monad dejned by Y and 9, Then there exists an isomorphism between (SH)O(in the sense of Section 2.3) and %, such that

SO

is commutative. Proof. The correspondence for the objects is clear because ( S P , ) O and A, are bijective for the object classes. For the morphisms

holds naturally in the objects X and Y in Bmby the Yoneda lemma. By definition, the morphisms between the objects X and Y in S H are exactly the morphisms of the &'-algebras (&'X,pX)and ( Z Y ,p Y ) and

136

3.

UNIVERSAL ALGEBRA

hence the morphisms of the free %-algebras 9 ( X ) = Moraa(X, -) and 9( Y) = Moram(Y, -). By definition Y , -1, MoQI,(X, -)) MorsJ Y , X ) LX Mori(M~r~(,(

is natural in the ‘&-objects (= S*-objects). Hence Moram(X, Y) Morw(X, Y) with V = S J p . Let f : X - t X ‘ and g : Y ’ + Y be morphisms in ‘illmand let f ’ and g’ be the corresponding morphisms in (Sal.)O,then the Yoneda lemma implies that Morg,(X’, Y’) MOrg,(f,g)

1

g

Moryo(X’,Y’) lMorygo(f’.g’)

Mora,(X, Y ) g Morypo(X,Y ) is commutative. So the compositions under this application of morphisms coincide. This clarifies the significance of the construction of Kleisli in Section 2.3, Theorem 1. Conversely, we now have a method at hand to reconstruct the algebraic theory from an algebraic category Funct,(%, S) and the corresponding forgetful functor. One has to restrict (9’#)O : So-+ (SH)O only to the full subcategory Noof So. With these means we can also show the significance of consistent algebraic theories.

THEOREM 2. Let A : No-+ % be an algebraic theory, Funct,(%, S ) the corresponding algebraic theory and X the monad defined by the monadic forgetful functor Y : Funct,(%, S ) -+ S . Then the following are equivalent: (1) A : No--t ‘ is? consistent. I (2) There exists an %-algebra A whose underlying set has more than one element. (3) The natural transformation E : Id, X is argumentwise a monomorphism . (4) X : S --t S is faithful. --f

Proof. (1) => (2): Since A is faithful, Mora(n, I) has at least n elements, the projections. But Mora(n, -) is the free algebra generated by n. (2) 5 (3): Let (A, a) be an X-algebra and let A have more than one element. Let X be an arbitrary set. Then there is an injective map i : X -+ A X . Since w ( A ): A -+ X A -+A is the identity on A the map € ( A is ) injective and hence also r(A)i. Since E is a natural transformation we get E(A)i = H ( i ) E(X).Thus c ( X ) : X --t X ( X ) is a monomorphism.

3.4

ALGEBRAIC FUNCTORS

137

(3) 3 (4): Let f , g : X - + Y be two maps in S with #f = #g. Since E ( Y )is a monomorphism and e(Y)f = # f e ( X ) , we get f = g Hence, &' is faithful. (4) * (1): 9'& is faithful because &' is (Section 2.3, Corollary). So (9'&)O restricted to No is faithful and consequently A also is.

Algebraic Functors

3.4

Let 'i!I be an algebraic theory, ?lr : Functn(21, S)-+ S the corresponding forgetful functor, and Z the corresponding monad.

LEMMA1. Let f : ( A ,a ) -+ ( B ,,9) be a morphism of #-algebras. Then on the set f(A) = C there exists exactly one #-algebra structure y : X C --t C, such that the factorization morphisms g : A --t C and h : C -+ B o f f are morphisms of X-algebras. Proof. We use the following commutative diagram:

I

A-C-B

8

P iI h

where hg = f , g is a surjective map, and h is an injective map, that is the factorization off through the image off. Since g is a retraction and h is a section (in S),X g and Z h is a factorization of X f through the image of X f . Let x and y be the factorization of SXf = f a through the image. Then there are maps y1 and y z making the above diagram commutative. But y = yzyl is the only morphism making both squares in the diagram commutative, since h is a monomorphism and #g is an epimorphism. If one uses the fact that g, Z g , and ##g are retractions, then the axioms for an algebra are easy to verify.

COROLLARY 1. Funct,(%, S) has epimorphic images. The resulting epimorphisms are surjectiue on the underlying sets.

138

3.

UNIVERSAL ALGEBRA

Proof. The corollary is implied by Lemma 1 of this section, Corollary 2 of Section 3.2, and Section 3.3, Theorem 1. Although Funct,(B, S) has epimorphic images, the example of Ri shows that Funct,(%, S)is not balanced in general. O n the other hand, a bijective morphism of %-algebras is an isomorphism because &' preserves isomorphisms. Let (A, a) be an &-algebra and X a subset of A. This defines a morphism 9'""(X) --t (A, a). Let (B, 8) be the image of this morphism. Then X C B C A and (B, 8) is the smallest subalgebra of (A, a) containing X.In fact, there is an %-homomorphism from 9'""(X) into each subalgebra of (A, a) containing X. (B, 8) is called the subalgebra of (A, a) generated by X . An %-algebra (A, a) is generated by the set X if X C A and if (A, a) coincides with the subalgebra of (A, a) generated by X.If X is finite, then (A, a) is said to be finiteZy generated. LEMMA2. by X.

There is onZy a set of nonisomorphic %-algebras generated

Proof. Let X C A and f : &'X+ A be a surjective map. Then on A there is at most one %-algebra structure a : %A + A such that f : Y " ( X ) + (A, a) is a homomorphism of algebras. In fact, in the diagram 2 2 X %2 A

%f is a surjective map. There is an 2-algebra structure on A if and only if (A, a) is generated by X.Since there is only a set of nonisomorphic surjective maps with domain %X the lemma is proved. COROLLARY 2. There is only a set of nonisomorphic %-algebras generated by epimorphic images of X .

Proof.

X

has only a set of nonisomorphic epimorphic images.

Let 33' : B + 8 be a morphism of algebraic theories. By composition 9' induces a functor

3.4

ALGEBRAIC FUNCTORS

139

called the algebraic functor. Furthermore, the diagram

S

is commutative, where F forgetful functors.

=

Funct,(g, S) and where the

Viare the

LEMMA 3. Let A E Funct,(%, S) and B E Funct,(B, S). Let f : A -+ 9-B be an %-homomorphism. Then there exists a minimal B-subalgebra B' of B such that there is an %-homomorphism g : A -+ 9-B' making the diagram

commutative. Proof. Let a2 = Funct,(%, S). T h e functor Mor,(A, 9--) : Funct,(B, S) -+ S preserves limits and Funct,(B, S) is locally small and complete. By the Lemma of Section 2.1 1, to each f : A -+ 9-B there is a minimal subobject B' C B and a morphism g : A -+ FB' such that the diagram becomes commutative.

THEOREM 1. Each algebraic functor is monadic. Proof. Let F , Vl , and V2be as in Lemma 3. Let fo , f l : A -+ B in Funct,(B, S) be F-contractible. Then fo , fl is Vl-contractible too because of Vl = V2F.There exists a difference cokernel g : 9 B -+ C of Ffo,Ffl in Funct,(%, S) if and only if there exists a difference cokernel h : Vl -+ X of Y29-f0, V 2 F f l in S. Then there exists also a difference cokernel k : B -+ D of fo ,f l in Funct,(23, S) and V f l k = V l k = h = V 2 g . Since V2generates the difference cokernels under consideration, we get F k = g. k is uniquely determined by V 2 g = h since Vl is monadic. Hence 9- generates difference cokernels of 9contractible pairs. By Section 2.3, Lemma 5 the functor V2generates isomorphisms. There is a uniquely determined morphism f : T I @ 9+1& 9-9 in Funct,(%, S) for a diagram 9 in Funct,(B, S) which is determined by

140

3.

UNIVERSAL ALGEBRA

the universal property of the limit. But V2f is an isomorphism since Yl = preserves limits. Hence f is an isomorphism. Consequently, F preserves limits. By Section 2.9, Theorem 2, it is sufficient to find solution sets for 9. Let A E Funct,(N, S ) and f : A 9-B be an N-homomorphism. By Lemma 3, the set given in Corollary 2 is a solution set of A with respect

Yx

-j

to

9-.

THEOREM 2. Let N be an algebraic theory. Then the functor Funct,(N, S ) + Funct(N, S ) defined by the embedding is monadic. Proof. It is sufficient to show that Funct,(%, S) is a reflexive subcategory of Funct(%, S) (Section 2.4, Theorem 2). By the construction of the limits in both categories (argumentwise) the embedding preserves limits. Let A E Funct(N, S) and B E Funct,(%, S ) . Let f : A + B be a natural transformation. Let B’ C B be the %-subalgebra of B generated by f ( A ( 1 ) ) . Let rp : n --+ 1 be an n-ary operation in %. Then the following diagram is commutative:

Here K(n) is uniquely defined by the fact that B’(n) is an n-fold product of B’( 1) with itself. For rp = pni the diagram is commutative by definition. In the general case we only have to prove the commutativity B‘(rp)k(n) = k(1) A(rp). But this holds because i(1) is an injective morphism. Thus, by Corollary 2 a solution set is given. COROLLARY 3. Funct,(%, S ) is cocomplete. Proof. Section 2.1 1, Theorem 3 and the dual of Section 2.7, Theorem 1 imply the corollary.

3.4

141

ALGEBRAIC FUNCTORS

Let 7 : 9 -F 53 be a functor. A morphism f : A + B is called a relatively split epimorphism iff is an epimorphism and Ff is a retraction. Dually, one defines a relatively split monomorphism. An object P E V is said to be relatively projective (relatively injective) if for all relatively split epimorphisms (monomorphisms)f in 2? the map Morw(P,f ) (MOT&, P)) is surjective. If F is the identity functor, then all objects are relatively projective and relatively injective (Section 1.10, Lemma 3). If 9 has a left adjoint functor 9, then Y D is relatively projective for all D E 9. Mor,(D, F j ) is surjective. In fact, Mor,(YD, f ) Let 7 be an algebraic functor with the left adjoint functor 9'.We say that the objects Y L ) are relatively free. Then each relatively free object is relatively projective. Since 9'- : Funct,(%, S ) S is also an algebraic functor, namely the functor induced by A : No + 2l, each free %-algebra is relatively projective with respect to the surjective %homomorphisms. I n this case we say the relatively projective objects are also %-projective. ---f

THEOREM 3 . Let 91 be an algebraic theory. Then there exists a finitely generated, %-projective generator in Funct,(Bl, S ) . Proof. T h e free %-algebra Mor,(l, -) has this property. T h e only thing to show is that Mor,( 1, -) is a generator. This assertion follows from Morf(Mor,( 1, -), A ) g ?+'-(A)and from the fact that ?+'- is faithful. Let (A, a) be an %-algebra. A congruence on (A, a ) is a kernel pair x , y : p -+ A in S such that (x, y ) : p + A x A defines a subalgebra ( p , n) of (A, a) x (A, a). Clearly, (x, y ) : p -+ A x A is injective since (x, y ) h = (x, y)k implies xh = xk and y h = y k and thus h = k by the uniqueness of the factorization morphism. Furthermore, TI is uniquely determined by the algebra structure on A x A.

LEMMA 4. Let (A, a) be an 21-algebra. x , y : p -+ A is a congruence on (A, a ) if and only if there is an algebra structure T : Xp + p on p such that x , y : ( p , T ) -+ ( A ,a) is a kernelpair in Funct,(A, S ) . Proof. Let x,y be a kernel pair in Funct,(2[,S). Since 9'-:Funct,(%,S) S preserves limits x , y is a kernel pair in S . Furthermore, (x, y ) : ( p , T ) + ( A , a) x (A, a) is a subalgebra since Funct,(%, S ) is complete. Now let x , y : p -+ A be a congruence. Since ( x , y ) is an %-homomorphism, also x = p,(x, y ) and y = p,(x, y ) are %-homomorphisms. Now let h : A -+C be a difference cokernel for x, y in S . Then there is a k : C --f A with hk = 1,. Then h l , = h = hkh for the pair of morphisms 1, , kh : A -P A. Thus there exists exactly one g : A -+ B with xg = 1, and y g = Kh and hence y g x = khx = Khy = y g y . So -+

142

3.

UNIVERSAL ALGEBRA

x, y : ( p , n) + ( A , a ) is a 9'--contractible pair. Consequently, there is an %-algebra structure on C such that (C, y ) is a difference cokernel of x, y in Funct,('N, S)(Section 2.3, Lemma 4 and Section 3.3, Theorem 1). By Section 2.6, Lemma 4, a kernel pair in Funct,(%, S)of ( A ,a ) -+ (C, y )

has p as underlying set up to an isomorphism. However, the %-algebra structure on p is uniquely determined by the injective morphism (x, y) : p + A x A. Hence x, y : (p, n ) -+ ( A , a ) is a kernel pair in Funct,(PI, S). We denote the difference cokernel of a congruence x, y : ( p , T)+ ( A ,a ) by (Alp, a ' ) or simply by A/p since the corresponding a-algebra structure is uniquely determined. A2 = A x A and A with the morphisms p , , p , : A x A --+ A and 1, , 1, : A + A are always congruences on ( A ,4.

COROLLARY 4. A n '$1-homomorphism f : ( A ,a ) -+ (C, y ) is a dzfference cokernel in Funct;(%, S)if and only iff : A -+ C is surjective. Proof. The proof of Lemma 4 implies that differences cokernels are surjective maps. Now let f : ( A , a ) -+ (C, y ) be an %-homomorphism with a surjective map f : A + C. Let x, y : ( p , n ) ( A ,a ) be a kernel pair off. Then x, y : p --f A is a kernel pair off in S.Since f : A -+ C is a difference cokernel for x, y in S we get that f : ( A , a ) (B, 18) is a difference cokernel for x, y in Funct,(a, S)as in the proof of Lemma 4. ---f

---f

THEOREM 4 (homomorphism theorem). Let x, y : p + A and x', y' : p' -+ A be congruences on (A,a). Let v : p -+ p' be given with x'v = x and y'a, = y ( p C p f ) . Let g : ( A ,a ) Alp be a dzflerence cokernel of x, y and h : ( A ,a ) + Alp' be a dzfference cokernel of x', y'. Then there is exactly one %-homomorphism f : A/p -+Alp' such that ---f

is commutative and f is surjective (as a set map). Proof. We have (x', y ' ) = ~ (x, y ) in S. Since (x', y ') is injective (x', y f ) vn = (XI,y ' ) m ' z a , implies qm = T ' S ~ that, is, p is an 2l-

homomorphism. Then the existence o f f follows from the properties of the difference cokernels. f is surjective because h is surjective.

3.4

143

ALGEBRAIC FUNCTORS

COROLLARY 5. Let f : (A, a) .--t (B, p) be an %-homomorphism and let x, y :(p, 7)--.f ( A ,a) be a kernelpair off. Then Alp e Im( f )as %-algebras.

Proof. The morphism (A, a! -+ Im( f)is surjective (Corollary 1) hence, a difference cokernel of its kernel pair (Corollary 4 and Section 2.6, Lemma 4). Since the kernel pairs of (A, a)-t Im( f)and f : (A, a)+(B, 18) coincide on the underlying sets, they coincide in Funct,(A, S). This implies the assertion.

LEMMA 5. Let A be afiberproduct of B and B' over C and let D be afiber product of E and E' over F. Let a morphism of diagrams (B, B', C) --t (E, E', F ) be given such that C -t F is a monomorphism. Then B x B' + E x E' and D + E x E' are uniquely dejined and A is a jiber product of B x B' and D over E x E . A

*B

E

Proof. G i v e n X - t B x B ' a n d X + D w i t h ( X + B ( X - t D + E x E'); then (X+B x B'-+B+C-tF)

=

+F

x B'+E x E')

=

(X+B x B'+B'+C+F).

Since C -t F is a monomorphism, we get ( X + B x B' -t B + C) = ( X + B x B' --t B' -t C). Thus there exists exactly one morphism X - t A with ( X - t A -+B x B') = (X-t B x B'). ( X + E - + F ) = (X -t E + F ) implies that there is exactly one morphism X -+ D with ( X - t D - 2 3 ) = ( X - t E ) and ( X - t D - E ' ) = ( X - E ' ) . But both the original morphism X -+ D and X + A + D have this property.

144

3.

UNIVERSAL ALGEBRA

Thus (X ---t D) = ( X +A + D) and A is a fiber product of B x B' and D over E x E .

THEOREM 5 (first isomorphism theorem). Let ( A , a ) be an %-algebra. Let i : ( B ,8) + ( A ,a ) be an 2l-subalgebra and let x, y : (p, T ) -+ ( A , a ) be a congruence on A. Let h : ( A ,a ) Alp be a diference cokernel of x, y. Let p(B) = h-'(hi(B)) in S. Then (1) p(B) is a subalgebra of ( A , a ) ; (2) p n B2 is a congruence on B ; ( 3 ) Blp n B2 g p(B)/p as %-algebras. Proof. hi(B) is an %-algebra as the image of hi (Lemma 1 and Corollary 1). p(B) = h-'(hi(B)) is an %-algebra as a limit of %-algebras (Section 3.2, Theorem). p n B2 = p n (B x B) is a kernel pair of hi : (B,8) --+ ( A , a ) + A/p, for --f

I - X'

(P n B2,4

(B,B)

''

1

(P,4 .

x

l i

(44

hi

h

A/p Ah

Y

is a special case of Lemma 5. Similarly, p A p(B2) is a kernel pair of

p(B) ---t ( A , a ) + A/p. Thus, B / p n B2

hi(B) E h(h-l(hi(B))

p(B)/pn P ( B ) ~

If a E p(B) and if a is p-equivalent to b, then also b E p(B), since a and b are mapped onto the same element in Alp. Thus, p(B) is saturated with respect to p. So we write p(B )/p instead of p(B)/p A p(B)2.

THEOREM 6 (second isomorphism theorem). Let q C p ( C A x A ) be congruences on A. Let p / q be the image of p Then p / q is a congruence on A/q and

=

---f

A

x

A + A/q

x

A/q.

Alp (A/q)/(p/q) Proof. Let r be the kernel pair of A/q + Alp. Then Alp g (A/q)/r by Corollary 4 and Theorem 4. A + A/p and A/q + Alp induce a morphism of kernel pairs p -+ r. By Lemma 5 , p-AxA

3.5

145

EXAMPLES OF ALGEBRAIC THEORIES AND FUNCTORS

is a fiber product in Funct,(%, S)and in S.Since the set-theoretic fiber product is {(a,6 ) E r x A x A If(.) = g(b)} and since g is surjective, p -+ r is also surjective. Hence, p -+ r -+ A / q x A / q is a decomposition of p --f A x A --f A / q x A / q through the image, thus r = p / q .

3.5 Examples of Algebraic Theories and Functors We know already some examples of algebraic categories namely S, S*, Gr, Ab, ,Mod and Ri. T o give more examples in a convenient manner we shall partly use the usual symbols (+, -, [,I, etc.) for the definition of the operations, and we shall represent the identities as equations between the elements of Mor,(n, 1). T h e reader will easily translate these data into the general formalism, if he compares them with the example of the algebraic theory of groups.

Examples 1.

M-(multiplicative) object: T h e algebraic theory of M-objects is defined by ( I ) a multiplication p : 2 --t 1 (2) without identities

2. Semigroup : (1) p : 2 - t 1 with p ( x , y ) = x y (2) ( v ) z = 4 Y . )

3. Monoid: (1) p : 2 -

(2) ox 4.

1; e : O + 1 with p ( x , y ) = x y ; e(wl) = 0 d; ( x y ) z = x ( y x )

=x =

H-( Hopf)object : (1) p : 2- 1; e : O-+ 1 (2) ox = x = x€)

with

p ( x , y ) = x y ; e(wl) = 0

5 . Quasigroup: ( I ) cu:2-+1;

,!?:241; y:2-+1 with + , y ) = x y ; B(.,y) = 4% Y ( X , Y ) = x\y ( 2 ) ( x / y ) y = x; +\y) = y ; X\(.Y) = y ; (.y)/y

These equations mean that the equation x y with respect to each of the three elements.

=z

=

x

is uniquely solvable

146

3.

UNIVERSAL ALGEBRA

6. Loop: (1) Quasigroup together with e : 0 ---t 1 and e(wl) (2) ox = x = xo

=

0

Here the operations and the identities of the quasigroup shall hold.

7.

Group: (1) p : 2 - t 1; s : 1 -+ 1 ; e : 0- 1 with p ( x , y ) = x y ; s(x) = x - l ; e(wl) = 1 (2) lx = x; x-1x = 1 ; (xy)z = x(yz)

8. Ring : ( 1 ) Group (p, s, e) together with v : 2 + 1 with p(x, y ) = x y ; s(x) = -3; e(wl) = 0; v(x, y ) = xy x; x ( y ). = ( x y ) (4 (x y ) z = (2) x y = y

+

+

(4+ ( Y 4

+

+

9. Unitary ring: ( 1 ) Ring together with e’ : 0 + 1 (2) lx = x = x l

+

with

+

e’(wl) =

1

10. Associative ring: (1) Ring together with (2) ( X Y b = X ( Y 4 11.

Commutative ring: ( I ) Ring together with (2) XY = YX

12. Anticommutative ring: ( 1 ) Ring together with (2) xx = 0 This identity implies x y = - y x . T h e converse does not hold in general.

13. Radical ring: (1) Associative ring together with g : 1 + 1 with g(x) = x’ (2) x x’ xx‘ = x x‘ x’x = 0

+ +

+ +

14. Lie ring: (1) Anticommutative ring (where we write v ( x , y ) = [ x , y ] instead of v(x, y ) = x y ) (2) [x, [ Y , 41 [ Y , rz, 211

+

+ [z, [ x , y I l = 0

3.5

147

EXAMPLES OF ALGEBRAIC THEORIES AND FUNCTORS

15. Jordan ring: (1) Commutative ring together with (2) ( ( x x ) y ) x = ( x x ) ( y x )

16. Alternative ring: (1) Ring together with (2) @X>Y = x(xY); 4 Y Y )

=

(XY)Y

17. R-module (for an associative ring): (1) Commutative group together with r : I --t I for all r E R (2) (r r’)m = rm r’m; r(m m’) = rm rm’; r(r’m) = (rr’)m

+

18.

+

+

+

Unitary R-module (for a unitary, associative ring R): (1) R-module together with (2) lm = m

19. Lie module (for a Lie ring R): ( I ) Commutative group together with r : 1 + 1 for all r E R r’)m = (rm) (r’m); [r, r’]m = (r(r’m)) - (r’(rm)); (2) ( r r(m m‘) = (rm) (rm’)

+

+

+

20.

+

Jordan module (for a Jordan ring R): (1) Commutative group together with r : 1 + 1 for all r E R m’) = (rm) (rm’); ( 2 ) (r + r’)m = (rm) + (r’m); r(m r(r’((rm) (rm)))= (rr‘)((rm) (rm)); r((rr)m)= (rr)(rm)

+

+

+

+

21. S-right-module (for an associative ring S) like an S-module, but (ss‘)m= s’(sm) holds instead of (ss’)m = ~ ( s ‘ m ) 22. R-S-bimodule: (1) R-module and S-module with the same commutative group with for all T E R and S E S (2) r(sm) = s(rm) 23.

k-algebra (with an associative, commutative, unitary ring k ) : (1) Ring together with r : 1 4 1 for all r E k (2) (Y r’)x = ( r x ) (r’x); r(x y ) = ( r x ) (ry); (rr’)x = r(r’x); Ix = x ; r(xy) = (rx)y = x(ry)

+

+

+

+

24. k-Lie-algebra, k- Jordan-algebra, and alternative k-algebra arise from Example 23 if we replace “ring” by “Lie ring,” “Jordan ring,” or “alternative ring,” respectively.

148

3.

UNIVERSAL ALGEBRA

25. Nilalgegra of degree n: (1) k-algebra together with (2) x" = 0 26. Nilpotent algebra of degree n: (1) k-algebra together with (2) xl(xz (-..x,) -..) = 0 It is interesting to know which algebraic structures are not equationally defined. In special cases it is easy to find properties of algebraic categories which do not hold in these cases. For example, the fields (with unitary ring homomorphisms) do not form an algebraic category because not each set-theoretic product of two fields can be considered as a field again (Section 3.2, Theorem). For the same reason, integral domains (with unitary ring homomorphisms) do not form an algebraic category (example of Section 2.12). T h e divisible abelian groups do not form an algebraic category because the monomorphisms are not always injective maps (Section 3.2, Corollary 2 and Section 1.5, Example 1). Morphisms of algebraic theories always define algebraic functors. Many universal constructions in algebra are left adjoint functors of algebraic functors. Most morphisms of algebraic theories are defined by adding operations and (or) identities, as we found already in the examples of algebraic theories. I n the following examples we shall not give special explanations if we use the above mentioned construction. '

Examples

27. % (= algebraic theory of groups) + 8 (= algebraic theory of commutative groups) induces an algebraic functor Funct,(B,

S) -+ Funct,('%, S)

T h e left adjoint functor is called the commutator factor group. 28.

% (= k-module) + 23 (= associative, unitary k-algebra) defines (as in Example 27) the functor tensor algebra.

29.

9I (= k-module) + B (= associative, commutative, k-algebra) defines the functor symmetric algebra.

unitary

30. % (= k-module) + B (= associative, anticommutative k-algebra) defines the functor exterior algebra. 3 1. % (= associative ring) + B (= associative, unitary ring) defines the functor adjunction of a unit.

3.6

ALGEBRAS I N ARBITRARY CATEGORIES

149

-+ b (= unitary, associative k-algebra), where the Lie-multiplication is mapped into the operation xy - y x with the associative multiplication, defines the functor universal enveloping algebra of a Lie algebra.

32. 91 (= k-Lie-algebra)

[,I

33. 91 (= k-Jordan-algebra)

b (= unitary, associative k-algebra), where the Jordan multiplication is mapped into the operation xy y x with the associative multiplication, defines the functor universal enveloping algebra of a Jordan-algebra. -+

+

34. 91 (= monoid)

-+ 23 (= unitary, associative functor monoid ring.

ring)

defines the

35. Let f : k -+ k’ be a unitary ring homomorphism of commutative, unitary, associative rings. 91 (= k-module or k-algebra) -+ 23 (= k’-module or k’-algebra

respectively) defines the functor base (-ring) extension.

36. 91 (= NO)-+ B (= unitary, associative (commutative) k-algebra) defines the functor (commutative) polynomial algebra.

3.6

Algebras in Arbitrary Categories

Let V be an arbitrary category and 91 an algebraic theory. An %-object in ?? is an object A E %? together with a functor d : ‘$20 -+ Funct,(BI, S), such that + d Funct,(PI,

s)

S is commutative with h, = Mor,(-, A). This means that each set Mor,(C, A ) carries the structure of an 91-algebra and that each morphism f : C -+ C’ induces an 21-homomorphism Morq(C’, A ) -+ Mor,(C, A). Here we meet again the common principle (see Section 1.5): Generalize notions from the category S to the category %? with the help of the bifunctor Mor,(-, -) in the covariant argument. One wants to carry out many computations and definitions for %-objects as for 91-algebras. But %-objects ( A ,d)have no elements in general. As a substitute we have the elements of the %-algebras Mor,(C, A), often denoted by A ( C ) (or better d ( C ) ) . Then one has to

3.

150

UNIVERSAL ALGEBRA

check in addition that the computations and definitions behave naturally with respect to C. An 2l-morphism f : ( A ,d) (B, 39) is a natural transformation f : A --+ B. This defines a natural transformation Vf : h, -+ h, , which again defines a morphism f * : A -+ B by the Yoneda lemma. T h e category of PI-objects and PI-morphisms will be denoted by 59%)and will be called category of %-objects in V . If 9 : PI -+ 23 is a morphism of algebraic theories, then this induces a functor W): Uc") -+ --f

THEOREM 1. Let U be a category with finite products. Then there is an equivalence U(") E Funct,(P[, U ) such that, for all morphisms 9 : B --+ 21 of algebraic theories, the diagram

is commutative.

Proof. Let ( A ,d)be an %-object. Then we can regard d as a bifunctor d : V0 x S with d ( C , n) g d ( C , I>"

=

Morv(C, A)" g Moryp(C, An)

and d ( C , v)

Moryp(C, A@): Morv(C, Am)

-

Moryp(C, A")

where Am : Am-+ An exists by the Yoneda lemma. Let f : ( A ,d)--t (B, 99)be an 2l-morphism and let f * : A -+ B be induced by f. Then, f (C, n ) g Mor,(C, (f* ) n ) . These applications define a functor -+ Funct,(%, U). Let X E Funct,(%, U). Then A = X(l) and &(C, n) = Mor,(C, An) define an object in 59%). In fact, let v : n --+ 1 be an n-ary operation in 2l, then we get X(p) : An -+ A , hence d ( C , v) = Mor,(C, X(rp)) : Mor,(C, An) -+ Morv(C, A). Given x : X -+XI in Funct,(2l, U ) we obtain Morq( -,

x( -))

: Mary( -, X ( -))

-

Moryp( -, X'( -))

and hence a morphism d -+ d'where a?'is determined by X'. This defines a functor Funct,(2l, U ) -+ W'). These two functors are, by construction, inverse to each other.

3.6

ALGEBRAS IN ARBITRARY CATEGORIES

151

With this construction it is easy to verify that 3 ' : b -+ 2I defines the commutative diagram in Theorem 1. A forgetful functor % from W') to V is defined by (A,&) t-t A and f F+ f *; then this forgetful functor, composed with the equivalence constructed in the proof, is the evaluation on the object I, hence 9'- : Funct,(%, %) + V. Now we show that product-preserving functors preserve %-objects and 2l-morphisms. This is stated more precisely in the following corollary.

COROLLARY 1 . Let %' and 9 be categories with jinite products. Let

S : V - + 9 be a product-preserving functor. Then there is a functor '3 : %Per) -+9 ' ) such that the diagram

is commutative. Proof.

Let '3'

=

Funct,(%, S). Then the diagram

is commutative for 99'-(X) = SX(1) = Y Y ( X ) and SY(x)= S X ( 1 ) = Y'3'(x). I n particular each representable functor Mor,(C, -) : V --t S preserves products, hence %-objects and %-morphisms. But this was the way %-objects and %-morphisms were defined. A co-%-object in V is an 2l-object in go.A co-%-morphism in %? is an %-morphism in Vo.

THEOREM 2. Let 2l be an algebraic theory. Then the free %-algebras in Funct,(S[, S ) are co-%-objects and the free %-homomorphisms are co- %-morphisms. Proof. Let X E S and A E Funct,(2l, S). Then M o r t ( 9 X , A ) LZ Mor,(X, Y A ) natural in X and A. But since A is an %-algebra,

3.

152

UNIVERSAL ALGEBRA

Mor,(X, Y A ) carries the structure of an %-algebra (namely the structure of A*). This again is natural in X and A. T h u s

that is, F X is a co-%-object in Funct,(%, S). Similarly, one proves the assertion for the co-Pt-morphisms. By a result of Kan, the free %-algebras and %-homomorphisms coincide with the co-PI-objects and co-2I-morphisms in Funct,(%, S) in the case of the algebraic theory of groups %, This assertion, however, does not hold for arbitrary algebraic theories. Let A : No+ % and B : No + 23 be algebraic theories. We define a tensor product Qt @ 23 of algebraic theories:

where L,(n) and LB(n) are the identities occurring in the representation of % and 23 by gB(%) and gB(23) respectively, and where va E Mor,(m, I), t,hB E MorB(r, I), #B x x # B MorB(n, ~ m), and yA x x vAE Mor,(n, r). All unions are disjoint unions. Then, in particular, morphisms PI 91 @ 23 and !I3 + 2l @ 23 of algebraic theories are given. ---f

THEOREM 3. Let %' be a category with Jinite products. Then there is an isomorphism

Proof.

By Section 1.14, Lemma 3 we have

Thereby, Funct,(%, Funct,(23,%')) is carried over into Funct,,,(Pt x 23,%'), the category of those bifunctors that preserve products in each argument separately. We define an isomorphism

3.6

153

ALGEBRAS IN ARBITRARY CATEGORIES

Given F E F ~ n c t , , ~ ( 2>:I 8, %) and G E Functn(21@ 8, U). Then 9 and 8 are determined by the following properties:

P(i,j ) = P(1,l)$i P(P,f)

%().

=NP,1 l ) ~ ( l l , P i )===wl,P")(CLi, =

1,)

Y(1)"

Y(T) = 9(1)1

We define

means rp E Im(2l --+ 2l 08) and with (p, p ) : ( i ,j ) -+( A , m). Here similarly for pB . The projections are assumed in Im(2l + 21 08 ) . We define, for natural transformation a : Pl 3 P2and /3 : + 8, ,

Thus, Q, and Yare functors. Furthermore, we have

Hence Q, and Y are isomorphisms.

COROLLARY 2. The tensor product of algebraic theories is commutative and associative up to isomorphisms.

154

3.

UNIVERSAL ALGEBRA

Proof. The algebraic theory is uniquely determined, up to isomorphisms, by the corresponding algebraic category and its forgetful functor. Since Funct,,,,,(% x 8,S) E Funct,,,,(d x 8,S)

we also have Funct,(8 @ 8, S)E Funct,,(B @ 8, S) and this isomorphism is compatible with the forgetful functors. Hence, 8 @ 8 8 @ a. The assertion about the associativity may be proved analogously.

LEMMA.Let ai : 0 - 1 ( i E I ) in 8 a n d & : 0 - 1 ( ~ E J in ) B begiuen, and let I and J be nonempty sets. Then the images of the ai's and pj's in 8 @ 8 are all equal. Proof. This is a consequence of $BpAr= pA+hBrn for

I =

m

=

0.

THEOREM 4. Given algebraic theories 8 with a : 0 + 1, p : 2 + 1 and p(aO1, 11) = l l = p(ll,aO1) and 8 with /3: 0- 1, v : 2- 1 and v(/301 , 11) = 1, = v( l1 ,POl). Then we get for the induced multiplications p* and v* in PI @ 23: (1) p* = v* (2) p * ( ~ pZ1) ~ ~=, p*, that is p* is commutative ( 3 ) p*(ll x p * ) = p*(p* x 11), that is p* is associative

Proof. Consider the commutative square v*

1 x 1 x x x-x 1 x 1

1

V*

li*xli*[

1

1

V*

1 x l - 1

Here the object in the left upper corner of the square is the object 4 = 1 x 1 x 1 x 1 in 8 @ 23. Then the square V'

Mor(n, 1) x Mor(n, 1) x Mor(n, 1)

x

X

Mor(n, 1)

x

Mor(n, 1)

.*x.,1

1)

v'

V'

X Mor(n,

1)

x Mor(n, 1)

1..

Mor(n, 1)

3.6

155

ALGEBRAS IN ARBITRARY CATEGORIES

is also commutative, where p'

=

Mor(n, p*) and v' = Mor(n, v*). Let

*

be an element in Mor(n, 4) and let p'(w, y) = w * y and v'(w, x) = w x. Then for all w, x, y , and z we have (w * y ) (x * z ) = (w * x) ( y * z). Since a* = /?*, let (n -+ 0 A 1) = 0 be the neutral element with respect to p' and also v'. Then we get

*

w*z =(w~o)*(o'z)=(w*o)'(o*z)=w'z

-

y x = (0 * y) . (x * 0)

w * (x z )

=

(w * 0) * (x * z )

=

(2)

(0 * x) * (y * 0) = x * y

= (w . x)

- (0

*

(1)

-

z ) = (w * x ) z

(3)

COROLLARY 3. Let 2l be the algebraic theory of groups and 8 the algebraic theory of commutativegroups. Then b % 0 @ 9I (n times)for n 2 2. Proof. 'ill 0% has exactly one neutral element and exactly one multiplication which is commutative. Thus at most the commutative groups may be group objects in Funct,('ill, S). But all commutative groups are group objects in Funct,(Q[, S), because MorGr(A,B) is a group, in case B is a commutative group. Hence, Funct,($, S) T h e assertion for n

Funct,(%, Funct,(%, S))

> 2 may be shown

analogously.

COROLLARY 4. The only group object in Ri is the zero ring { 0 >. Proof. All multiplications and neutral elements coincide. Thus for a group object in Ri we get 0 = 1 and 0 = 0 * a = 1 a = a for all a of the group object. Let 'ill be the algebraic theory of groups. If U is the category Top, then Funct,(Bl, U ) is called the category of topological groups. If U is the category of analytic varieties, then Funct,(Bl, U ) is called the category of analytic groups. If " 0 is the category of finitely generated, unitary, associative, commutative k-algebras and k a field, then Funct,(%, U ) is called the category of ajine algebraic groups. Let Sn be the n-sphere in Htp* = (if?. T h e homotopy groups of a pointed topological space T are defined by n,( T) := Mor,(Sn, T ) . These sets have a group structure which is natural in T . Thus the n-spheres are co-group-objects in Htp*.

3.

156

UNIVERSAL ALGEBRA

Problems 3.1.

Show that the following categories are not algebraic categories:

(a) the torsionfree abelian groups (an abelian group G is called torsionfree, if ng implies n = 0 or g = 0 for all n E w and g E G); (b) the finite abelian groups.

=

0

3.2. Let % be an algebraic theory. Let X E S and A E Funct,(%, S). Let A be generated by X and let f : X + A ( l ) be an arbitrary map. I f f can be extended to an %-homomorphism g : A + A , then g is uniquely determined by f .

3.3. Let ‘ube an algebraic theory. Then there is an %-algebra A for which A(1) consists of exactly one element. All %-algebras with one element are isomorphic. 3.4. Under with conditions on the algebraic theory ‘udoes there exist an empty %-algebra ? 3.5. Let % + B be a morphism of algebraic theories, Y : Funct,(B, S) + Funct,(%, S) the corresponding algebraic functor, and Y : Funct,,(’u, S) -+ Funct,(B, S) the left adjoint functor of Y. Let X E S , f X the %-algebras freely generated by X , and E E Funct,(B, S ) . T h e coproduct B a ( X )of Y B and 9 X is called a generalized polynomial . mapf : X + B(1) may algebra of B with the variables X. We have X C B q [ ( X ) ( l )Each uniquely be extended to an %-homomorphism B a ( X ) --+ S ( B ) such that the restriction to YE is the identity and to X is the map f. This morphism is called the insertion homomorphism. Let % be the algebraic theory of unitary, associative rings, 23 the algebraic theory of unitary, associative, commutative rings. Describe the insertion homomorphism.

3.6. Let R and S be in Ri. L e t J : R -+ S be a unitary ring homomorphism. Show thatf induces a morphism from the algebraic theory of unitary R-modules to the algebraic theory of unitary S-modules. Describe the corresponding algebraic functor Y and its left adjoint functor. What is the meaning of the assertion that the corresponding algebraic functor 9 is monadic [Section 2.3, Theorem 21 ? Has Y a right adjoint functor ? 3.7. Show that polynomial algebras, tensor algebras, and symmetric algebras are co-monoid-objects in the category of associative, unitary (commutative) k-algebras (see Section 3.5).

3.8. Let k be a field. The polynomial algebra k [ X Jin one variable (generated by one element) and the monoid algebra k [ Z ] generated by the additive group of integers H (Section 3.5, Example 34 for algebraic functors) are cocommutative co-group-objects (co-%-objects with the algebraic theory % of commutative groups) in the category of unitary, associative, commutative k-algebras. T h e coproduct in this category is the tensor product. Describe the comultiplications k [ X ] k [ X ] 0k [ X ] and k [ Z ] + k [ Z ] 0 k [ Z ] . (Determine the value of 0 E X = { 0 ) and of 1 E Z under these maps.) --f

3.9. Let V be a category with finite products, % an algebraic theory, and 93 a small category. Characterize the %-objects in Funct(93, U) as “pointwise” %-objects in Y: such that morphisms in D induce %-homomorphisms. 3.10. Use Section 2.1 1, Theorem 4, Section 2.4, Theorem 2, Section 2.3, Theorem 2, the proposition of Section 3.3, and the following remarks to prove the following theorem of Birkhoff:

PROBLEMS Let K be a full subcategory of Funct,(91, S) with

(I)

Z contains a noncmpty 91-algebra;

(2) (3) (4)

Z is closed with respect to subalgehras; Y is closed \\itti respect to products; 'X is closed with respect to images of 91-homomorphisms with domain in V.

Then C is an algebraic category.

157

4 Ab elian Categories Up to now the theory of abelian categories is by far the best developed. The notion stems from a paper of Grothendieck in 1957. Many important theorems, which may be found for module categories in many textbooks, will be proved here more generally for abelian categories. A great deal may be represented in a much nicer and simpler way by these meansfor cxamplc, the theorems on simple and semisimple rings, where we shall use the Morita theorems. T h e desire to preserve also the computations with elements (similar to the computations for modules) leads to the embedding theorems. T h e proof of these theorems uses mainly methods developed by Gabriel. For example, the construction of the 0th right-derived functor originates from the paper of Gabriel listed in the bibliography.

4.1 Additive Categories Let 59 be a category with a zero object, finite coproducts, and finite products. We saw in Chapter 1 that %?is a category with zero morphisms which are uniquely determined. Let finite index sets I and J and objects A, with i E I and B, with j E J in %? be given. Furthermore, let a familyfij : A , -+ B j of morphisms in V for all i E I and j E J be given. T h e coproduct of the Ai will be denoted by 1l. A, and the injections by q, : A, + A, . Similarly, we denote the product of the Bi by Bi and the projections by p , : Bj -+ Bi. Then there are uniquely determined morphisms f, : A, -+ Bi with p,f, = f,, and a uniquely determined morphism f : 1l. A, -+ Bj with pifqi = fij . If, in particular, the morphisms 6 , : A, + A j are given for all i, j E I with Sit = l A Eand Si, = 0 for i # j , then the morphisms uniquely determined hereby will be denoted by 6, : A, + A,. Correspondingly, we define 6, : 1l. Bi-+ B, . For a family of morphisms g, : A, + B, for all i E I there exists exactly

n

n

n

n

n

158

n

4.1

159

ADDITIVE CATEGORIES

one morphism U gi : Ai -+ Bi with JJg i q k = q k g k for all k €1. Furthermore, there is exactly one morphism ngi : Ai -+ B, with p , gi = g,p, for all k E I. But then the square

n

n

is commutative because the morphism from H Ai to the morphisms g, if j = k fjk = 0 if j f k

n

n Bi is induced by

n

In fact, fjk = P k 6 B giqj = p k gisAqj * Let A , : A + A, with A, = A and piAA = I, be the diagonal and let V, : A, A with V,qi = 1, be the codiagonal (see Section I . 1 1). Now assume that 6 is an isomorphism for all finite products or coproducts respectively. Then we take for the products-for example, of the (A&-the coproducts, that is, A, ; the projections arise from the composition of the original projections with 6, that is, pi6, : Ai -+ Ai . Thus we get 6 = 1, that is, we may identify finite products and finite coproducts. T h e coproduct of finitely many A i will 0A, and will be then also be denoted by @A, or by A, @ A , @ called a direct sum. We shall treat the morphisms similarly. In fact, by the above considerations finite products and finite coproducts of morphisms also coincide. A category V is called additive category if

n

--f

( 1 ) there exists a zero object in Y, (2) there exist finite products and finite coproducts in V, ( 3 ) the morphism 6 from finite coproducts to finite products is an isomorphism, and (4) to each object A in 9 there exists a morphism sA : A that the diagram

is commutative.

-+A

such

4.

160

ABELIAN CATEGORIES

Let 9? be an additive category. On the morphism sets Mor,(A, B ) we define a composition written as addition by

f + g := ' E ( f

-

for all f, g E Mor,(A, B). Furthermore, we define a morphism t, : A @ A A @ A by plt,ql = pzt,q, = 0 and plt,q, = pzt,ql = 1A . Then t,A, = A , by definition of the diagonal and dually V B t B= V, . Thus we get

f +g

'E(f@g)

= 'EtB(.f

@ g ) tAd,4 == ' B ( g

Of)

=g

+f

that is, the addition is commutative. T h e associativity of the addition follows from the commutativity of the diagram

A

7

II?

II)

in fact ( A , @ 1) A , as well as (1 @ A,) A , is the diagonal. One verifies componentwise ( f @ 0) q1 = (f@ 0) A , and dually p,(f @ 0) = V,( f @ 0), hence f 0 = p,( f @ O)q, = f . Because of ( f @ g ) ( h @ h) = ( f h O g h ) and A,h = ( h @ h ) A , we get

+

(f + g)

',(fog) 0gh) = f A + gh Dually we get h ( f + g ) = hf + hg. These equations together with the =

= vB(fh

forth condition for additive categories show that the sets Mor,(A, B ) with the given addition form abelian groups and that the composition of morphisms is bilinear with respect to this addition.

THEOREM. Y is an additive category i f and only i f there exists a zero object in Y , if there exist finite coproducts in V and if each of the morphisms sets Mor,( A, B ) carries the structure of an abelian group such that the composition of morphisms is bilinear with respect to the addition of these groups. Proof. We saw already in the preceeding considerations that an additive category Y has the properties given in the theorem.

4.1

161

ADDITIVE CATEGORIES

Now assume that these properties hold for V. First we show that the finite coproducts are also finite products. Let A , ,..., A, be objects in +? and let JJ Ai be their coproduct. T h e morphisms S i j : A, + Ai with Sii l,, and Sii = 0 for i # j define for each j exactly one morphism pi : JJ Ai + A j with PjQi

(1)

= aii

Furthermore, we get from

for all j

=

l,.,., n the relation

Zlere we used that the zero morphism is the neutral element for the group structure of Mor,(A, B). In fact 0 = O(1, 1,) = 01, 01, = 0 0. Now let morphisms fi : C + Ai be given. Then 1 qifi : C -+ JJ A i is the desired morphism into the product for pi 1 q i f i = f j . If Ai is another morphism with pig = fi, then g : C --t

+

+

+

T h e n by (2) we have

that is, JJ Ai together with the projections pi is a product of the Ai . T h e morphism 6 : JJ A i Ai is defined by pj6qi = Sij . But since pjlllAZqi = Sij by (3),we get 6 = 111,, . T h u s also point (3) of the definition of additive catcgories holds. As in the beginning of this section, a finite family of morphisms f i i : Ai --f Bi defines exactly one morphism f : @Ai -+ @ B j with p j f q i = fij . We also write the morphism f as a matrix f = (fij). Let another family of morphisms g j p : Bj-+ C, be given. Let h = (gjk)(fij). Then ---f

P&,

=PL(R,~)

n

C q j P , ( f i i ) qz j

1i

gjnfij

4.

162

ABELIAN CATEGORIES

Hence the composition of morphisms between direct sums is similar to the multiplication

Using this matrix notation we get A,

+

Hence f g for s, = -1,.

=

=

V,( f @ g) A , . In particular we get AA(1, This completes the proof.

0sA) A ,

=

0

COROLLARY 1. Let V be an additive category. Then there is exactly one way to defne an abelian group structure on the morphism sets such that the composition of morphisms in V is bilinear.

+

Proof. We saw that f g = V,(f @ g ) A , must hold. T h u s the addition can only depend on the choice of the representatives of the direct sums. T h e universality of the definition of V, ,f @g, and A , shows that the addition is unique. The assertion made in Corollary 1 is the main reason for the fact that we did not use the properties that are characteristic for an additive category by the theorem for the definition of an additive category. If we consider Mor,(A, B ) as an abelian group in the following, then we shall also write Hom,(A, B). COROLLARY 2. Let V be an additive category. Let A , ,..., A, and S be objects in V and let q, : A, + S and p i : S + A, for i = 1, ..., n be morphisms in V. The following are equivalent: (a) S is a direct sum of the A, with the injections qi and theprojectionsp,. (b) piqi = aij for all i and j and qipi = Is .

Proof. If S is a direct sum of the A , , then (b) holds because of (1) and (2). Assume that (b) holds. As in the proof of the theorem we then see that S together with the projections pi is a product of the Ai . Dually, we get that S is a coproduct of the A, with the injections qi . Observe that the dual of an additive category is again an additive category because all four properties used in the definition are self-dual.

4.2

163

ABELIAN CATEGORIES

I n an additive category V the endomorphisms of an object A, that is, the elements of Hom,(A, A ) , form an associative ring with unit, the so-called endomorphism ring.

Example 1 T h e category Ab of abelian groups is an additive category. I n Chapter 1 we saw that Ab has a zero object and products. Let f , g E Mor,,(A, B ) . Then ( f + g ) ( a ) := f ( a ) g ( a ) defines a group structure on Mor,,(A, B ) which satisfies the conditions of the theorem.

+

Example 2 T h e category of divisible abelian groups with all group homomorphisms as the morphisms is an additive category. Here we define the addition of morphisms as in Example 1. T h e only thing to show is that there are finite coproducts. I t is sufficient to show that finite coproducts in Ab of divisible abelian groups are again divisible. Let A and B be divisible, that is, n A = A and n B =: B for all n E N, then n ( A @ B ) = n A @ n B =

A OB. 4.2 Abelian Categories In this section let %? be an additive category. Furthermore, assume that each morphism in %? has a kernel and a cokernel. Let two morphisms f , g E Hom,(A, B ) be given, and let h = f - g . We want to show that the kernel of h coincides with the difference kernel o f f and g. Given c : C -+ A with f c ==gc, then hc = f c - gc = 0; thus there exists exactly one d : C + Ker(h) with c = (C + Ker(h) --+ A). Furthermore, (Ker(h) -+ A f B ) = (Ker(h)

--f

A

B)

Dually, the cokernel of h also coincides with the difference cokernel off and g. Thus there are difference kernels and difference cokernels in 9.

LEMMAI . Let 59 be an additive category with kernels. Then %?is a category with jinite limits. Proof. Since V is a category with difference kernels and finite products, we can apply Section 2.6, Proposition 2.

164

4.

ABELIAN CATEGORIES

Let f : A -+ B be a morphism in 'X. I n the diagram A

there is exactly one morphismg with q'g = f becausep'f = 0. We denote Ker(p') also by KerCok(f). Dually,f may be uniquely factored through CokKer( f ). Both assertions may be combined in the commutative diagram 9

Ker(f) +A

-

-lf

P

CokKer(f)

Ih

9' Cok(f) P' B t-KerCok(f)

where h is uniquely determined by f.I n fact the morphismg may uniquely be factored through CokKer( f ) because of 0 = f q = q'gq, hence gq = 0. By Section 1.9, Lemma 1 both q and q' are monomorphisms and p and p' are epimorphisms. If h' instead of h also makes the diagram commutative, then q'hp = q'h'p, hence h = h'. An additive category with kernels and cokernels, where for each morphism f the uniquely determined morphism h : CokKer( f ) -+ KerCok( f ) is an isomorphism, is called an abelian category. Example An important and well-known example for an abelian category is the category ,Mod of unitary R-modules. As in Section 4.1, Example 1, one shows that ,Mod is an additive category. I n the theorem of Section 3.2 and in Section 3.4,Corollary 3 we saw that there are kernels and cokernels in ,Mod. T h e assertion that h : CokKer( f ) -+ KerCok( f ) is an isomorphism is nothing else than the homomorphism theorem for R-modules. One of the aims of the theory of abelian categories is to generalize theorems known for .Mod to abelian categories. This will be done in the following sections. Since there are no elements in the objects of a category, the proof will often be more difficult and different from the proofs for ,Mod. T o prevent these difficulties we shall prove metatheorems at the end of this chapter which transfer certain theorems known for .Mod without any further proof to arbitrary abelian categories.

4.2

ABELIAN CATEGORIES

165

Now let %' be an abelian category for the rest of this chapter unless we ask explicitly for other properties for %.

LEMMA 2. (a) Each monomorphism in V is a kernel of its cokernel. (b) Each epimorphism in V; is a cokernel of its kernel. (c) A morphism f in V is an isomorphism if and only i f f is a monomorpkism and an epzmorplzism.

Proof. (a) Let f be a monomorphism and let f g = 0. Then g = 0. Thus g may uniquely be factored through 0 -+ D(f ) (= domain( f )), i.e., Ker( f ) = 0. T h e cokernel of this zero morphism is 1 : D( f ) -+ D(f ). T h e commutative diagram 0 Cok(f)

W) 2W )

+

-If - lh

R(f)

t-

KerCok(f)

implies that D(f ) and KerCok( f ) are equivalent subobjects of R(f ) (= range( f 1). (b) follows from (a) because the definition of an abelian category is self-dual. (c) In (a) we saw that the kernel of a monomorphism is zero. Similarly, the cokernel of an epimorphism is zero. Then (c) follows from the commutative diagram 0

-

W) 2W )

LEMMA 3. For each morphism f in %? the image o f f is KerCok( f ) and the coimage o f f is CokKer( f ). Proof. A morphism f may be factored through KerCok( f ). Since there are fiber products in V, $? is a category with finite intersections. Let A be a subobject of R(f ) through which f may be factored, then f may be factored through A n KerCok( f ). Since D( f ) -+ KerCok( f ) is an epimorphism, A n KerCok( f ) -+ KerCok( f ) is an epimorphism and a monomorphism, hence an isomorphism by Lemma 2. T h u s D(f ) + A

I66

4. ABELIAN

CATEGORIES

may also be factored through KerCok( f ). Dually, one gets the proof for the coimage. Because of Lemma 3, we shall always write I m ( f ) instead of KerCok( f ) and Coim( f ) instead of CokKer( f ).

A morphism f : A COROLLARY. I m ( f ) = B.

-+

B is an epimorphism

if

and only

if

Proof. By Lemma 2,fis an epimorphism if and only if B = CokKer(f). KerCok(f) = I m ( f ) , the morphism f is an epiBy CokKer(f) morphism if and only if the subobject Im( f ) of B coincides with B.

4.3 Exact Sequences A sequence (fl , f,)of two morphisms in an abelian category $7 f f2 A , -L A, + A,

is called exact or exact in A, if Ker(f,) A sequence ***

f . A,,, Ai L

=

Im(fl) as subobjects of A , .

fi+l

+Ai+B

* a .

of morphisms in V is called exact if it is exact in each of the Ai+l, that is, if Ker(fi+,) = Im(fi) as subobjects of Ai+l . If the sequence is finite to the left side or to the right side, then this condition is empty for the last object. An exact sequence of the form

is called a short exact sequence. Let f : A -+ B be a morphism in %?.Then B -+ Cok(f) is an epimorphism. By Section 4.2, Lemma 2 we then get ( B + Cok(f))

=

( B -+ CokKerCok(f))

If Ker(fi+J = Im( f ), then Cok(fi) = CokKerCok(f,) = CokIm(fi) = CokKer(fi+l) = Coim(fitl). Hence the definition of exactness is self-dual.

LEMMA 1.

The sequence A

for the morphisms ( A -+ B

-+

A B %C C)

=

is exact if and only if we have 0 and (Ker(g) -+ B -+ Cok( f )) = 0.

4.3 Proof.

Let A

-+ B

-+

167

EXACT SEQUENCES

C be exact. Then we have trivially ( A -+ B -+ C ) = 0

that is, Im( f ) C Ker(g). Furthermore, we obtain an Coim(g) -+ Cok( f ) through which B -+ Cok( f ) may be (Ker(g) -+ B -+ Coim(g)) = 0. If (A -+ B -+ C) = 0, then Im( f ) C Ker(g). If, (Ker(g) -+ B + Cok( f )) = 0, then Ker(g) -+ B may through KerCok( f ) = Im( f ), hence Ker(g) _C Im( f ). A sequence

epimorphism factored. But furthermore, be factored

fi = 0 for all i is called a complex. Obviously this notion is selfwith fi+l dual. LEMMA 2. B is a monomorphism. (a) 0 + A -+ B is exact if and only if A (b) 0 -+ A -+ B -+ C is exact if and only if A + B is the kernel of B C. 0 -+ A -+ B -+ C -+ 0 is exact if and only if A -+ B is the kernel (c) of B -+ C and if B -+ C is an epimorphism. ---f

--f

Proof. (a) B y the corollary of Section 4.2, A -+ B is a monomorphism if and only if Coim(A -+ B ) = A = Cok(0 -+ A). (b) If A -+ B is the kernel of B -+ C, then Im(A -+ B) = ImKer(B -+ C) = Ker(B -+ C). Furthermore, A -+ B is a monomorphism. T h e converse is trivial. (c) arises from (b) and the assertion dual to (a).

LEMMA 3. Let % be an abelian category. Let A, , A, , and S be objects in % and let qi : Ai --+ S and pi : S + Ai (i = 1, 2 ) be morphisms in %. The following are equivalent: ( 1 ) S is a direct sum of the Ai with the injections qi and the projections pi . (2) piqi = lA,for i =I 1, 2 and the sequences

o

--

A,

P s2 A , +o

and

o are exact.

P

--+

A , 42_ s A A , -+

o

4.

168

ABELIAN CATEGORIES

(3) q1 and q, are monomorphisms, p , and p , are epimorphisms, and we have 41P1+ q2P2 = 1s and (q1PJ2 = 41P1 * Proof. (1) 3 (2): By Section 4.1, Corollary 2 it is sufficient to show the exactness of 0 - A , -+ S + A,+O

p , is an epimorphism because of p2q2 = 1. Given f : B + S with

+

p,f = 0, then f = (qlpl q2p2)f = q l p l f , i.e., f may be factored through 9,. This factorization is unique since q1 is a monomorphism. (2) => (1): Let fi : B + Ai be given. Let f = qlfl q 2 f 2 . Then p i f = f i . If a morphism g : B -+ S satisfies the condition pig = fi , then pi(g -f) = 0. Hence g - f may be factored through A , , that is, g - f = q,h. Then g - f = qlplqlh = qlpl(g -f) = 0. (1) 3 (3): By Section 4.1, Corollary 2, assertion (3) is trivially implied by (1). If ( 3 ) holds, then qlplqlpl = qlp, = q l l A l p l . By cancellation of the monomorphism q1 and the epimorphism p , we obtain PlQl = ]A1 . (1 - q1P1)2 = 1 - Q l P l implies (q2P2I2 = qzP2 , hence p2q2 = lA, . Furthermore, we have

+

P192

= PlQlP142P292= Pl(q,Pl)(l - 41Pd 9 2 = Pl(qlP1 - (!71Pd2) 9 2 = 0

and analogously p2ql = 0. Then (1) holds by Section 4.1, Corollary 2. Let f be an endomorphism of S withf = f . fmay be factored through the image off. Let p , : S -+ Im( f ) and q1 : Im( f ) -+ S. If we factor 1 - f = q 2 p 2 , then S = Im( f ) @ I m ( l -f). But by (2) we get Im(1 -f ) = Ker( f ) and hence, S = Im( f ) @ Ker(f).

LEMMA 4. (a)

The commutative diagram

d

B-+C is a jiber product if and only if the sequence

O-PLA@B%C with f =

is exact.

(3

and

g

= (c,

--d)

4.3

EXACT SEQUENCES

169

( b ) Let the commutative diagram in (a) be a jiber product. The morphism c ; A -t C is a monomorphism if and only if b : P -+ B is a monomorphism. (c) Let the commutative diagram in (a) be aJiberproduct. If c : A -+ C is an epimorphism, then the diagram is also a cojiber product and b : P - t B is an epimorphism.

Proof. (a) We define

f

=:

(3

and

g

= (c,

--d)

T h e minus sign, of course, could stand before any of the other morphisms a, b, or c because the only reason for it is to achievegf = 0. If the diagram in (a) is a fiber product and h : D -+ A @ B is given with gh = 0, then

Thus there exists exactly one morphism e : D + P with ae = h, and be = h,, that is, with f e = h. Conversely, each pair of morphisms h, : D -+ A and h , : D -+ B with ch, = dh, hence with g h = 0, defines exactly one morphism e : D -+ P with fe = h , i.e., with ae = h, and be = h , . (b) If c : A -+ Cis a monomorphism, then by Section 2.7, Corollary 5 b : P-+ B is also a monomorphism. Now let b : P+ B be a monomorphism. Let ( D -t A -+ C ) = 0. If we set ( D -+ B ) = 0 then there exists exactly one morphism D -+ P with (D-+ A ) = (D-+ P -+ A ) and ( D + P-+ B ) = 0. Since P -+ B is a monomorphism, we get (D+ P ) = 0 and hence (D-+ A ) = 0. This means that A -+ C is a monomorphism. (c) If c : A --t C is an epimorphism, then c = ( A -+ A @ B -+ C) is an epimorphism, hence also A @ B -+ C. By Lemma 2, the sequence 0 -+ P - + A @ B -+ C -+ 0 is exact. By (a) the diagram in (a) is a cofiber product. T h e assertion dual to (b) implies (c). In the following we shall denote the cokernel of a monomorphism by B / A . This corresponds to the usual notation for R-modules. I n the dual case we shall not introduce any particular notation for the kernel of an epimorphism. T h e applications which assign to each subobject of an object B a quotient object and to each quotient object a subobject are inverse to each other. Furthermore, they invert the order if, in the class

170

4.

of subobjects, we set A

ABELIAN CATEGORIES

< A' if and only if there is a morphism a such that A

A'

is commutative, and if, in the class of quotient objects, we set C if and only if there is a morphism c such that

< C'

C'

C

is commutative. This follows from the commutative diagram with exact rows O-A-B-C'---+O

0 +A'-

B

--

0

C

where a exists if and only if c exists.

LEMMA 5. In an abelian category V there exist finite intersections and finite unions of subobjects. The lattice of subobjects is antiisomorphic to the lattice of quotient objects of an object. Proof. Since V has fiber products, there exist finite intersections in V . Let A and B be subobjects of C. Then we define A U B = Im(A @ B + C). In fact, let D be a subobject of C' and let morphisms C --+ C', A -+ D,and B --+ D be given such that the diagrams A+C

D

B-+C

-

C'

D-

C'

4.3

171

EXACT SEQUENCES

are commutative. Then there exists a morphism A (A@B+C-,C')

0B

--f

D such that

=(A@B+D-tC')

Hence, Im(A 0B + C ) + C + C' may be factored through D + C'. Thus the class of subobjects of V is a lattice. T h e preceeding considerations imply immediately the second assertion of the Lemma.

If there exist infinite products in the abelian category $9, then there exist arbitrary intersections of subobjects in the category V. If there exist infinite coproducts in V, then there exist arbitrary unions of subobjects in the category V. COROLLARY.

Proof. If 9 has infinite products, then V is complete and thus there exist arbitrary intersections of subobjects. If V has infinite coproducts, then the proof of Lemma 5 may be repeated verbally for infinitely many subobjects.

LEMMA 6. (a) Let f : A -+ B and g : B -+ C be morphisms in an abelian category %. Then Im(gf) C Im(g). (b) Let f, g : A

-+

B be morphisms in V. Then

Wf + 8)c W f )" Im(g)* Proof. (a) T h e diagram

Wh)

is commutative, A + Im( f )-+ Im(h) is an epimorphism, and Im(h) -+ Im(g) ---f B is a monomorphism. Hence Im(h) = Im(gf) C Im(g). (b) We have j+-g

=

(

A

~ @AA - ~ - t r n ( j ) @ I r n ( g ) ~ B O B ~ B )

By definition, Im(f) u Im(g) Im(f g) Im( f 1 u W g ) .

+ c

=

Im(Vb). Hence, by (a), we get

4.

172

ABELIAN CATEGORIES

4.4 Isomorphism Theorems

THEOREM (3 x 3 lemma). Let the diagram 0

0

-

1 - 1 1 -0

0-

A , 4A ,

0

B,

1 _+

0

A,

-1 -1 B,

0

B,

1 1 . 1 c, 1 1 1 0

c 2

c 3

0

0

be commutative with exact rows and columns. Then there are uniquely defined morphisms C , + C , and C,-+ C, making the above diagram commutative. Furthermore, the sequence 0 -+ C1-+ C , -+ C, + 0 is exact.

Proof. T h e existence and uniqueness of C, + C , and C,-+ C, is implied by the facts that C, = Cok(A, -+ B,) and ( A , -+ C,) = 0 and, respectively, C , = Cok(A,-+B,) and (A,+ C,) = 0. Furthermore, C , + C, is an epimorphism because (B, -+ C,

-+

C,)

=

(B, -+ B,

-+

C,)

is an epimorphism. If we omit in the diagram the object C , and the morphisms B, -+ C, and C, + C , , then the remaining diagram is selfdual. Furthermore, the sequence 0 -+ A, -+ B,

--f

C,+ C, -+ 0

(1)

is exact. For reasons of duality, it is sufficient to prove the exactness of 0 + A, -+ B , -+ C, , that is, A, = Ker(B, -+ C.J. Let D -+B, with ( D -+ B, -+ C,) = 0 be given. Then there exists D -+ A, with

(D+ B, -+ B,)

= (D+

A, + B,)

Since ( D - B , ) = 0 and A,+ B, is a monomorphism, we have ( D -+ A, + A,) = 0, hence there is a morphism D -+ A, with ( D + A,) = ( D -+ A, -+ A,). Since B, + B, is a monomorphism and (D+B1+B2)

= (D-tA,+B,-+B,)

4.4

I73

ISOMORPHISM THEOREMS

we have (D-+ B,) == (D4A, + B,). T h e uniqueness of this factorization follows from the fact that A, -+ B, is a monomorphism. We have Ker(B, --+ C,) = A, and Cok(B, -+ C,) = C , . T h u s C, = Coim(B, -+ C,) -= Im(B, -+ C,) = Ker(C, -+ C,) as subobjects of C, and C, + C, is an epimorphism.

COROLLARY 1 (first isomorphism theorem). Given subobjects A C/B. Then we have BIA C C/A and (C/A)/(B/A)

C B C C.

Proof. Apply the 3 x 3 lemma to the diagram 0

1 0 -+A

0

0

-A

I

-

+0

0

1

1

1

1

1

BIA

CIA

C/B

1

1

0

0

0

COROLLARY 2 (second isomorphism theorem). Given subobjects A C C A/(A n B), that is, the diagram and B C C . Then we have (A u B)/B 0

0

0

0

0

0

1

1

B

B / ( An B )

- -- 1 -1 - 1 - - -A nB

A --

A / ( An B )

AUB

(AU B)/A--+O

( A u B)/B

0

0

0

1

0

is commutative with exact rows and columns.

0

0

1 74

4.

ABELIAN CATEGORIES

Proof. T o apply the 3 x 3 lemma we have to show that B / ( An B ) ( A u B)/B is a monomorphism. Let D (D+B+A

-+

B with

-

u B - + ( A u B ) / A )= 0

be given. Then there is exactly one morphism D -+ A with

(D+ A -t A u B ) = ( D - t B-+ A u B ) Thus there is exactly one D -+ A n B with (D-+B) = ( D + A n B + B )

and

(D+A) =(D-+AnB-+A)

that is, A n B is the kernel of B -+ ( A u B ) / A .But the morphism CokKer(B -+ ( A u B ) / A )-+ ( A u B ) / A is always a monomorphism. Now let us apply the 3 x 3 lemma to show that

c, = ( ( Au ~ ) / A ) I ( B I (nA B ) ) vanishes. We have ( A + A U B+ C,) = 0 and ( B - t A u B - t C,) = 0. Thus by the definition of a union ( A u B -+C,) = 0. T h e diagram implies that A u B -+ C, is an epimorphism. Hence, C, = 0.

COROLLARY 3. Let C = A u B and A n B = 0. Then C is the direct sum of A and B with injectioas the embeddings of A and B into C . Proof. Insert A n B = 0 into the diagram of Corollary 2. Then A + A / ( An B ) + ( A U B ) / B and B -+ B / ( A n B ) + ( A u B ) / A are isomorphisms. If we take as projections foi the direct sum the inverses of these isomorphisms composed with A u B + ( A u B ) / Band A u B 3 ( A U B ) / A ,then we can easily apply Section 4.3, Lemma 3.

4.5 The Jordan-Holder Theorem An object A # 0 in an abelian category %' is called simple if for each subobject B of A either B = 0 or B = A holds. Let 0 = B, C B, C C R, = A be a sequence of subobjects of A which are all different. Such a sequence is called a composition series if the objects B,/B,-, are simple for all i = 1 , ..., n. T h e objects Bi/Bi-, are called factors of the composition series and n is called length of the composition series.

4.5

THE JORDAN-HOLDER THEOREM

175

LEMMA1. Let A C C and B C C be nonequivalent subobjects of C. Let CIA and CIB be simple. Then C = A u B. Proof. A C A u B and B _C A u B imply that at least one of the subobjects, for example B, is different from A u B . By the 3 x 3 lemma there is a commutative diagram with exact rows and columns 0

0

0

1

1

1 1

1 0

u

0-

AU B

1 1

4

(A u B)/B-

0

11 - 1u 10 1 0

C

C/(AuB)+O

C/B

C/(A B)-0

By hypothesis, we have ( A V B ) / B # 0 and ( A u B ) / B C CIB. Since CIB is simple, we get C / ( Av B ) = 0 hence C = A u B .

LEMMA 2. Let 0 = B, (C C B, = A be a composition series. Let C C A and let AIC be simple. Then there exists a composition series of A through C of length n: 0

=

C,C***CC,_,CCCA

Proof. T h e proof is by complete induction with respect to n. For n = 1, the only composition series of A (up to equivalence of subobjects) is 0 C A. Assume that the lemma holds for composition series of length n - 1. Consider the diagram

176

4.

ABELIAN CATEGORIES

where we may assume that C and Bn-l are nonequivalent subobjects of A , since otherwise there exists already a composition series through C. Thus by Lemma 1 we have A = C u B n - l . By the second isomorphism theorem Bn-,/(C n Bn-,) = A / C is simple. Since BnPlhas a composition series of length n - 1, there exists a composition series of Bn-l through C n B,-, of length n - 1. Hence, C n Bn-, has a composition series of length n - 2. This may be extended through C and A , for C / ( C n B,-l) = A/B,-, and A / C are simple.

THEOREM 1 (Jordan-Holder).

Assume that the object A in V has a composition series. Then all composition series of A have the same length and isomorphic factors up to the order. Proof. By complete induction with respect to the length of a composition series of minimal length of A. For n = 1, there exists only one composition series of A , as above. Assume that the theorem is already proved for all A with composition series of length < n - 1. Let two C B, = A and 0 = C, C C C,, = A composition series 0 = B, C be given. We form * * a

**.

B,,-2

Bn-1

Cm-2

'G,-,

Since, by the second isomorphism theorem, all factors of the diagram are simple A/Bn-, r Cm-I/(Bn-ln Cm-d

and

A/C,,,-, e Bn-l/(Bn-l n C,,,J

all sequences in the above diagram are composition series because the theorem holds already for B n P l . Here we used that B,-, and Cm-l are nonequivalent subobjects, for otherwise the assertion may be reduced to B n p 1 .Since Bn-, and Cn1-, have composition series of equal length, namely through Bn-l n Cnl-, , we get m = n. T h e factors of the composition series of Bn-l and CnLPldiffer only in Bn-,/(Bn-, n Cnl-l) and Cm-l/(Bn-ln CniPl).But both factors appear in the composition series of A through n C,,-,. Hence both given compos'tion series of A have the same length and isomorphic factors up to the order. If A has a composition series of length n, then we also say that the

4.5

177

THE JORDAN-HOLDER THEOREM

object A has length n. If A has a composition series, which by definition is finite, then we also say that A is an object of finite length.

PROPOSITION 1 . Let A be an object of jinite length and let C be a subobject of A. Then there exists a composition series of A in which C appears as a n element.

Proof. Let 0 = B,C We form the sequences

C B,

A be a composition series of A.

=

O=CnB,C...CCnBB,=C

and C

=

C U B,C

.*.C C U B,

=A

As in the proof of the second isomorphism theorem, one shows with the 3 x 3 lemma that the diagram 0

0

0

-

C n BiP1--

-

Bi-,/C

1 1n

C

0

0

In

1

Bi

-+

C n Bi/C n Bi-l

--+

0

1 1 B,-l

1

-

Bi/C n Bi -+ C u Bi/C u BiP1

4

1

1

1

0

0

0

0

is commutative with exact rows and columns. I n fact, we have C n BiP1= ( C n B i ) n BiPl . Furthermore, using both isomorphism theorems we obtain C u B J C u BzPlE (C u Bi/C)/(Cu Bi-,/C)

(B,/C n Bi)/(B1-JCn

Since B,/B,-, is simple, each factor object of B,/B,_, is either simple or 0, since the kernel of the morphism into the factor object is either 0 or simple. Hence just one of the objects C n B,/C n Bi-l or C u B J C uBi-, is simple and the other one is 0. If one connects the sequences given above, and if one drops all of the members which appear several times, except one, then this new sequence is a composition series through C.

178

4.

ABELIAN CATEGORIES

An object of finite length may well have infinitely many nonequivalent subobjects (see Problem 8). But by Proposition 1 each proper subobject has a length smaller than the length of the object. Hence in each set of proper subobjects of an object of finite length, the subobjects of maximal length are maximal, and the subobjects of minimal length are minimal, and such subobjects always exist if the given set is nonempty.

COROLLARY 1. An object has finite length if and only noetherian.

if it is artinian and

Proof. The only thing we have to prove is that each artinian and noetherian object A has finite length. In the class of subobjects of A , which are not equivalent to A , there is a maximal subobject B, . Since B, is again artinian and noetherian, we may construct B, , B, ,..., in the same way. This defines a descending sequence of subobjects of A. Since A is artinian, this sequence stops after finitely many steps. Furthermore, the factors of this sequence are simple by construction, hence this is a composition series of A.

COROLLARY 2. Let B be an object of finite length, and let the sequence 0 -+ A + B C -+ 0 be exact. Then A and C are objects of finite length, and we have --f

length(B)

=

length(d)

+ length(C)

I n particular, an epimorphism between objects of equal length is an isomorphism.

Proof. Let 0 = B, C C Bi = A C C B, = B be a composition series of B through A . Then ( B k / A ) / ( B k p 1 / A )B,/Bk-, is simple for all i < k n. Hence, 0 = B,/A C C B,/A = C is a composition series of length n - i. Furthermore, A has length i. T h e second assertion follows from the fact that the kernel of the epimorphism has length 0, and that each object of length 0 is a zero object.


a with (G’, a , a*) 2 (G’, a, /3) for all G‘ C G and all?! , > a. Since it is sufficient to show that 1. a cofinal subsequence becomes constant, we may assume that a* = a Let y be the first ordinal which has larger cardinality than the set of subobjects of G. y is a limit and we have B, = l i z B, for all a < y . If we consider the B, as subobjects of then B, = (J,. B, . Now f : G + By,, is a morphism which cannot be factored through B, . Such a morphisms exists as long as B, # , which we want to assumc now. We get a chain of subobjectsf-l(B,) of G and by Section 4.7, Corollary 4 we havef-’(B,) = f -l(B,). Let P I ) ;

+

u,,,

K

= {a

If-’(Ba)

Lf-l(&+1N

4.9

INJECTIVE A N D PROJECTIVE OBJECTS A N D HULLS

20 I

and let I K I be the cardinal number of K . Then I K I < I y I by the assumption on y . Furthermore, j a 1 < 1 y 1. By Lemma 2 of the appendix, there exists a p < y with a < fl for all a E K , that is, for all p' > /3 we have f -l(BB,)= f -I(Ba), hence f -l(B,) = f -I(B,). Since by our construction p* = p 1 we get ( f -l(B,), 8, y ) = ( f -'(Ba), /3, y 1). T h e morphism f ' :f -I(Bp)-+ B, induced by f can already be extended to a morphismg' : G Bysuch that the diagram

+

+

f-'(Ba)

-

BB -+

-+

G

B,

is commutative. Let g : G be the morphism induced by g'. Then g f f , but (g - f ) ( f - Y B J ) = ( g - f )(f-'(Ba>) = 0. Since B, is large in B v t l , we have Im(g -f) n B, # 0 ; hence there exists a morphism h' : G + (g - f )-' (Im(g - f ) n By)such that --f

(G

-

(g - f)-'(Im(g - f) n B Y )

-

W g -f) n BY) f 0

Let h : G -+G be the morphism induced by h'. Then (g - f ) h # 0 and Im((g - f ) h ) C B, . Since Im(gh) C Im(g) C B y , we have Im(fh) C B, , that is, Im(h) C f - I ( B , ) . Then, however, ( g - f ) h = 0 must hold. This is a contradiction to our assumption that By # By+, . In this proof we did not use all objects of the category %? to test the maximal essential extension, but only the generator G and the subobjects of G. Consequently, it is also sufficient to test the injectivity of objects only for the subobjects of G.

COROLLARY 2. Let V be a Grothendieck category with a generator G. Let Q E V be an object such that for all subobjects G' C G the map Hom,(G, Q) + Homr6(C',Q) is surjective, then Q is injective.

Proof. If Q has no proper essential extension, then Q is injective by Lemma 4. Let Q A be a proper monomorphism. Then there exists a morphism f : G -+ A which cannot be factored through Q. We form the commutative diagram -+

f-'(Q)--G

202

4.

ABELIAN CATEGORIES

By hypothesis there exists G -+ Q with

Let g = ( G + Q + A). Then g # f . As in the last paragraph of the preceeding proof, we then get Im(g - f ) n Q = 0. Hence, Q -+ A cannot be an essential monomorphism. With the present means we can now show that the Krull-RemakSchmidt-Azumaya theorem can also be applied to injective objects, similar to the case of objects of finite length that we proved in Section 4.8, Lemma 3. In fact, the difficulty is always to show that the endomorphism ring of certain indecomposable objects is local.

THEOREM 2. Let %? be a Grothendieck category with a generator. An injective object Q E %7 is indecomposable i f and only i f Hom,(Q, Q) is local. Proof. By Section 4.8, Lemma 3 we need only show one direction. Let Q be indecomposable and injective. Each monomorphism f : Q +Q is an isomorphism because f is a section and Q is indecomposable. Furthermore, each nonzero subobject of Q is large. In fact, let 0 # A C Q be given and let Q' be the injective hull of A. By Theorem l(5) we get Q' C Q. Hence we get Q' = Q because Q is indecomposable, that is, Q is an injective hull of A. T h e nonunits of Hom,(Q, Q) are the morphisms with kernel different from zero. If f, g E Hom,(Q, Q) with nonzero kernels are given, then Ker(f g) 2 Ker( f ) n Ker(g) # 0 by Section 2.8, Lemma I and because all nonzero subobjects of Q are large. Hence f g is a nonunit. If an injective object is given as a coproduct of indecomposable objects which then are necessarily also injective because they are all direct factors, then this representation is unique in the sense of the Krull-Remak-Schmidt-Azumaya theorem. Conversely, however, not each coproduct of injective objects is injective. Thus it will be of interest to know under which conditions we can decompose each injective object into a coproduct of indecomposable objects and when each coproduct of indecomposable injective objects is injective. We observe that each module category is a Grothendieck category and possesses a generator, namely the ring R. Thus all theorems proved in this section are also valid in module categories. Another important application of Theorem 1 will be used later on, namely the existence of injective cogenerators in a Grothendieck category with a generator. So we prove now the following more general theorem.

+

+

4.9

INJECTIVE A N D PROJECTIVE OBJECTS A N D HULLS

203

THEOREM 3. Let 5f be an abelian category with a generator G in which to each object there exists an injective extension. If V is complete or cocomplete, then there exists an injective cogenerator in %?. Proof. We prove the theorem for the case that %? has coproducts. I n case of the existence of products one may replace the coproducts by products everywhere in the proof. Since G has onlp a set of (normal) subobjects (Section 2.10, Lemma I), G has only a set of quotient objects G'. Let H be the coproduct of all these quotient objects and let K be an injective extension of H . We want to show that K is a cogenerator. Let f : A -+ B in C be given with f # 0. Then there exists a morphism G + A such that (G-+ A + B ) # 0. Let G -+ G' + B be the factorization of this morphism through the image. Then G' # 0 is a quotient object of G. Since the injection G' + N is a monomorphism, there exists a monomorphism (G'+ F Z 4 K ) # 0,

hence also (G -+ G' --+ Ii -+ K ) # 0. Since K is injective and G' + B is a monomorphism, there exists a morphism B + K such that the diagram G-G-+H

is commutative. (G --+ K ) # 0 implies also ( A 4K ) # 0. This proves that K is a cogenerator.

COROLLARY 3. Let W be a Grothendieck category with a generator. Then V has an injective cogenerator.

Proof. T h e corollary is implied by Theorems I and 3. COROLLARY 4. Let .Mod be a module category and 9.I be the set of maximal ideals M of R. Then each injective extension of RIM and RIM respectively is an injective cogenerator.

nMEgIUI

uMEm

Proof. If we observe that R is a generator in .Mod, then in comparison with the construction of the injective cogenerator in the proof of Theorem 3, we see that in the coproduct and product there are fewer factors. Rut since in a ring R each ideal I is contained in a maximal ideal M (see Appendix, Zorn's lemma), cach nonzero quotient module of R

204

4.

ABELIAN CATEGORIES

may be epimorphically mapped onto a module of the form RIM. Hence, we extend the diagram in the proof of Theorem 3 to a commutative diagram R R' --+ R / M H

-

1

1 1 A+B

P

K

where H is the coproduct or the product of the R I M and K is an injective extension of H. T h e rnorphism R 4 K is different from zero, thus the proof of Theorem 3 can be transferred to this case.

5 (Watts). Let .Mod and ,Mod be module categories. Let COROLLARY F : .Mod + ,Mod be a functor. F preserves limits if and only there exists an R-S-bimodule ,AS such that F ,Hom,(,A,, -), that is, if F is representable. Proof. If F is representable, then the assertion is clear. Assume that F preserves limits. By Corollary 4 and Section 2.1 1, Theorem 2 F has a left adjoint functor *F.Then 9-B Hom,(S, F B ) Horn,( * F S , B ) natural in B, hence F is representable. Here * F S has by definition the structure of an R-left-module. For s E S the right multiplication of S with s is an S-left-homomorphism r(s). Hence * F ( r ( s ) ) defines the structure of an R-S-bimodule on *FS.

4.10 Finitely Generated Objects Let %? be a category with unions. An object A E %? is called finitely generated if for each chain of proper subobjects {A,} of A also U Ai is a proper subobject of A. An object A E %? is called compact if for each family of subobjects {Ai} of A with (J Ai = A , there is a finite number A, ,..., A, of subobjects in this family such that A, u ... u A , = A.

THEOREM 1. A n object A compact.

E%

is finitely generated

if and only if it is

Proof. Let A be compact. Let {A,} be a chain of subobjects of A, with U Ai = A. Then there exist A, ,..., A, in this chain with A, u * . * U A, = A. One of these, e.g., A , , is the largest. Hence A = A , , and A is finitely generated. Let A be finitely generated. Let !B be a set of subobjects of A that is closed with respect to unions, and which contains A. Let ? be I l la subset of 8 that contains all elements except A. Since A is finitely generated,

4.10

205

FINITELY GENERATED OBJECTS

3 and 211 fulfill the hypotheses of Section 4.7, Lemma 1. If (J Ai = A for objects Ai E 93, then there exist finitely many A , ,..., A, with A , u u A, = A , that is, A is compact. With this thcorcni an algebraic notion (finitely generated) and a topological notion (compact) are sct in relation with each other. Here we have to remark that the usual definition in algebra of finitely generated objects is given with elements (Section 3.4 and Exercise 14), but that for proofs only the condition of the definition given here is used. This condition also admits easily the application of the Grothendieck condition.

COROLLARY 1. Let A be a module over a ring R. A is finitely generated in the algebraic sense if and only i f A is finitely generated in the categorical sense.

+

+ +

Ran , that is, if A is finitely generHa, Proof. If A = R a , ated in the algebraic sense, and if {Ai}is a chain of submodules of A with (J Ai = A , then, for each a j , there exists an A, with a j e A,. Let I = max(k), then ai E A , for all j = 1 ,..., n , hence A = A , . Now let A be finitely generated in the categorical sense, then A is compact. Let { a i } be a generating system for A , that is, A = (J R a t , then A = R a , u u Ra,, for suitable a , ,..., a , . Hence, A is finitely generated in the algebraic sense. Let 9 be again an abclian cocomplete category.

LEMMA 1. Let f :A -+ B be an epimorphism in %. If A is finitely generated, then B is also finitely generated.

u

Let { B i } be a chain of subobjects of B with Bi= B. Let Ai = f - ] ( B i ) . Then f ( U Ai)= f ( ( Jf-I(Bi)) = (J Bi = B. Since f is an epimorphism and the kernel of f is contained in (J A i , we get (J Ai = A , which may easily be seen by the 3 x 3 lemma. Furthermore, BiC Bi implies A i C A j , that is, {Ai}is a chain of subobjects of A. Since A is finitely generated, we get A, = A for some i. But B, = f ( A i ) = f ( A ) = B, hence B is finitely generated. An object A E % is said to be transfinitely generated if there is a set of finitely generated subobjects A i in A such that (J A , = A. Proof.

LEMMA 2. If’ V has a finitely-generated generator, then each object is transfinitely generated.

Proof. Let A E 9. Since by Section 2.10, Lemma 2 for each proper subobject A’ C A there is a morphism G + A , which cannot be factored

206

4. ABELIAN

CATEGORIES

through A', the morphism G -+ A which is induced by all morphisms of Hom,(G, A ) is an eimorphism, where we use in the coproduct as many objects as HorrS(G, A ) has elements. I n fact, the image must coincide with A. Hence A = (J A' where the A' are the images of the morphisms G + A. Since G is finitely generated, also the A' are finitely generated by Lemma I . Hence, A is transfinitely generated.

THEOREM 2. Let %? be a Grothendieck category. Let A E V be transfinitely generated. Then A is a direct limit of jinitely generated subobjects. Proof. We shall show that the union of finitely many finitely generated subobjects of A is again finitely generated. If then A = U A i and for each finite subset E of the index set A, = UiEEA i, then these (finitely generated) A, form a directed family of subobjects of A and we have A = UA,. Let B and C be finitely generated subobjects of A. Let {Di} be a chain of subobjects of B u C with Di= B u C. Then we have

u

(u

L),) n C = C

and

jUDi)nB=B

By the Grothendieck condition, we then get lJ(Di n C) = C and U (Din B ) = B. Since B and C are finitely generated there is a j with Din C = C and Din B = B, that is, Di2 B and D j 2 C. Hence, Di= B u C and B u C is finitely generated. By induction one shows that all finite unions of finitely generated subobjects are finitely generated. LEMMA3 . Let %? be a Grothendieck category and A E V be finitely Then there exist generated. Let f : A + IJ Bi be a morphism in %?. B, ,..., B, such that f may be factored through B, @ @ B, IJ Bi . ---f

Proof. Let B = IJ Bi and let for each finite subset E of the index set BE = BiGE Bi. Then the BE form a directed family of subobjects of B and we have B = U B E . Let A, = f-'(B,). Then A = f - ' ( B ) = f - l ( U BE) = Uf-'(B,) = U A,, Since A is compact, we get A = AE1u u AEr. Hence,

f(A)= f(AE1)u *..

U f ( A E r )C B E 1 U ..* U BE,. _C

B,

= B,

@

@ B,

If we compare the definition of a noetherian object with the definition of a finitely generated object, then it becomes clear that each noetherian

object must be finitely generated. T h e converse does not necessarily hold. A Grothendieck category with a noetherian generator will be called locally noetherian. A module category over a noetherian ring R (that is, R

4.10

207

FINITELY GENERATED OBJECTS

is noetherian in .Mod) is locally noetherian. We want to investigate some of the properties of the locally noetherian categories.

THEOREM 3. ( a ) In a locally noetherian category the coproduct of injective objects is injective.

(b) Let V be a Grothendieck category in which all objects are transfinitely generated and in uihich each coproduct of injective objects is injective. Then each finitely generated object is noetherian.

Proof. (a) Let G E V: be a noetherian generator and let {Qi} be a family of injective objects in 55'. Let G' C G be a subobject of G. Since G is noetherian, G' is noetherian, hence finitely generated. Let a morphism f : G' -+ M Qibe given which we want to extend to G. Then f may be @ Q n by Lemma 3. This direct sum is factored through Q1 @ injective as a product of injective objects. Hence the morphism G' 4 Q1 @ @ Q,, may be extended to G. Thus also f may be extended to G. Hence by Section 4.9, Corollary 2, Qi is injective. Let B be a finitely generated object in %?.T o prove that B is (b) noetherian it is sufficient to show that each ascending chain A , C A, C of subobjects of B becomes constant. Let A = U Ai and Qi be an injective hull of A/Ai . The morphisms A + A/Ai -+ Qi define a morphism A -+ . Since A is transfinitely generated, A = Cj with finitely generated subobjects Cj . We have Cj = Ai)n Cj = (J (Ai n Cj). Since Ci is finitely generated, we get Cj = Aion Cj for some io , Hence C j _C Ai for all i 3 io , that is, (Ci-+ A -+QJ = 0 for io. Thus Ci A + may be factored through all i Q1 @ ... @ Qi,. Hence each morphism Ci-+ A + Qi may be factored Qi . Since A = Cj , the morphism A -+nQi may be through factored through Qi . By hypothesis Qi is injective. Hence, A --t nQi may be extended to B:

nQi

(u

nQi

-+

u

u

n

A-B

LIQi

-n

Ql

Since B is finitely generated, B + IJ Qi may be factored through a direct sum Q1 @ ... @Qa . Then the same also holds for A and we get (A-lIQi)

=

(A-+QIO.*.@Qn-~Qi)

208

4.

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Thus, for almost all i, the morphism ( A -+ Q,) = ( A -.+ This means that, for almost all i, we have A = A,.

n Qi

-+

Q,) = 0.

COROLLARY 2. In a locally noetherian category all jinitely generated objects are noetherian.

Proof.

By Lemma 2, all objects in W are transfinitely generated.

COROLLARY 3 . Let R be a ring. R is noetherian if and only if the coproduct of injective modules in .Mod is injective.

Proof. If R is noetherian, then .Mod is locally noetherian. If, conversely, each coproduct of injective modules is injective, then R is noetherian as a finitely generated object.

LEMMA 4. Let % be a locally noetherian category. Then each injective object contains an indecomposable injective subobject. Proof. An object A E V is called coirreducible if for subobjects B , C C A with B n C = 0 we always have B = 0 or C = 0. If A is coirreducible then the injective hull Q(A)is indecomposable. In fact, let Q(A) = Q' @ Q", then Q' n Q" = 0 = (Q' n A ) n (Q" n A). Hence, Q' n A = 0 or Q" n A = 0. Since A is large in @ A ) ,we get Q' = 0 or

Q" = 0. Let Q E % be an injective object and let Q # 0. Since Q is transfinitely generated, Q contains a nonzero finitely generated subobject A. Since W is locally noetherian, A is noetherian. If A is not coirreducible, then there exist nonzero subobjects A, and B, of A with A , n B, = 0. If A, is not coirreducible, then there exist nonzero subobjects A , and B, of A , with A, n B, = 0. By continuing this process we get an ascending chain B, C B, @ B, C of subobjects of A. This sequence must become constant since A is noetherian. Hence, by this construction after finitely many steps, we must get a nonzero coirredicible subobject A' of Q. T h e injective hull of A' is again a subobject of Q and is indecomposable by the above remarks. With these means and the Krull-Remak-Schmidt-Azumaya theorem we now can make assertions about the structure of injective objects in locally noetherian categories. Here we refer again to Section 4.9, Theorem 2 and the remarks we made after this theorem.

THEOREM 4 (Matlis). Let V be a locally noetherian category. Each injective object Q in W may be decomposed into a coproduct of indecomposable

4.10

209

FINITELY GENERATED OBJECTS

injective objects Q = uia,Qi . If Q = Uje,Qi' is another decomposition into indecomposable injective objects Qi', then there exists a bijection q~ : I -+ J such that Qi g Qi(i)for all i E I . Proof. I t is sufficient to show the first assertion. T h e second assertion is implied by the theorem of Section 4.8,and Theorem 2 of Section 4.9. Since there is a generator in V ,Q has only a set of subobjects. We consider families {Qi} of indecomposable injective subobjects of Q with the property that (JQi = Qi as subobjects of Q. By Zorn's lemma, there Qi . Since Q' is an injective exists a maximal family {Qi}. Let Q' = subobject of Q, by Theorem 3 we have Q = Q'@ Q". If Q" 1s ' nonzero, then Q" contains an indecomposable injective subobject Q* and {Qi} u {Q"} fulfills the conditions for the families of subobjects defined Hence, Q = Q' = IJ Qi . above in contradiction to the maximality of {Qi}.

u

THEOREM 5 (exchange theorem). Let V be a locally noetherian category, {Qi}iE,a famiLy of indecomposable injective objects in %? and Q' an injective Q i . Then there is a subset K C I such that subobject of Q = U i e K Qi

OQ'

=z

Q.

Proof. Consider the subset J _C I with the property thatQ' n IJi,Qi = 0. Among these there is a maximal subset K by Zorn's lemma. Then K is an injective subobject of Q. So for all Qi , we have Q" = Q' @ U i E Qi Q" n Qi # 0. Since the Qi are indecomposable injectives, they are the injective hull of Ai = Q" n Qi . We want to show that Q is the injective hull of Q" and hence Q = Q". First Qil @ Q i z is an essential extension of Ail u Aiz , I n fact, if B # 0 is a subobject, then the image of B under f : Qil @ Qiz -+ Qil or g : Qil @ Qiz +Qiz is different from zero. Let B' # 0 be the image of B in Qil . Then B' n Ail # 0. In Section 2.8, Lemma 2 the morphism g and hence also f -I(D) n C -+ f ( C ) n D are epimorphisms. Thus B, = B nf -l(Ai1) # 0. If g(B,) # 0, then B, n g-'(Aip) # 0. Then B n ( A i l u Aiz) = B n f - l ( A i 1 )n g-l(Aiz) # 0. But if g(B,) = 0, then B, C Qil and B, n Ail # 0. Hence B n(At1

U At2)

1B nf-'(At,) n Ax1 # 0

By induction one shows that direct sums Q, of indecomposable injective objects Qi are an essential extension of a finite union A , of the Ai with the same index set. T h e A , and the Q, form directed families of subobjects of Q. We have Q" = Q" n Qi) 2 (Q" n Qi) = Ai = UA,.LetCfObeasubobjectofQ= uQ,.Then((JQ,)r\C= U(Q, n C) = C, henceQ, n C # OforsomeE. SowegetA, n C # 0,

(u

u

u

210

4.

(u A,)

But Q" n C 2 subobject of Q.

nC

ABELIAN CATEGORIES =

u (A, n C) # 0 means that Q" is a large

By Corollary 2 the last two theorems hold in each module category over a noetherian ring.

4.1 1 Module Categories In this section we want to characterize the abelian categories equivalent to module categories. Since we shall determine simultaneously the equivalences between module categories, we shall obtain a general view of these equivalences. I n this connection we shall prove the Morita theorems, which we shall apply in the next section for the discussion of the Wedderburn theorems for semisimple and simple rings. A projective object P i n an abelian category is calledfinite if the functor Homw(P, -) preserves coproducts. LEMMA1. Each finite projective object P in V is finitely generated.

If

$? is a Grothendieck category, then each finitely generated projective object

is finite.

u

Proof. Let {Pi} be a chain of subobjects of P with Pi = P. Then Pi -+ P is an epimorphism, hence there exists a morphism p : P-t 1l Pi w i t h ( P + U Pi-+ P ) = 1,. But p E Hom,(P, H Pi) g Hom,(P, Pi)has the form p = p , *.. p , . Thus we have also (P+P,@...@P,+P) = l,.ThusUr=lPi = P.SincetheP,form a chain, we get P = Pi for some i. Let 5f be a Grothendieck category. Then each morphismf : P -+ A, may be factored through a finite subsum A , @ * - . @ A, by Section 4.10, Lemma 3 because P is finitely generated and projective. f induces a A,) Hom,(P, A,). Since morphism g : P + A, in Hom,(P, A, -+ A, and 5f is a Grothendieck category, the morphisms At) are monomorphisms. Because Hom,(P, U Ai)--+ Hom,(P, Ai) Hom,(P, A,), we may regard f as an element of Hom,(P, Hom,(P, Ai).Since f can be factored through A , @ @ A,, p i g : P -+ A , is nonzero only for finitely many i, that is, f has in Hom,(P, A,) only finitely many nonzero components. Thus f lies in Hom,(P, Ai) of Hom,(P, A,). Conversely, each the subgroup element of Hom,(P, Ai) considered as a morphism from P into A , may be factored through a direct sum of (finitely many) Ais, hence lies in Homy(P, A,). This proves that the isomorphism

+ +

n

n

n n n

n

n

n

n

n

n

4.1 1

21 1

MODULE CATEGORIES

n

n

Hom,(P, Ai)E Hom,(P, A$) induces an isomorphism of the subgroups 1l Hom,(P, Ai)e Hom,(P, 1l Ai). A finite projective generator is called a progenerator. Now we can characterize the module categories among the abelian categories (up to equivalence).

THEOREM 1. Let V be an abelian category. There exists an equivalence .% : V + Mod, between V and a category of right modules ;f and only if% contains a progenerator P and arbitrary coproducts of copies of P. If 9is an equivalence, then P may be chosen such that Hom,(P, P ) .% Hom,(P, -).

R and

Proof. Let P be a progenerator in V. Then Hom,(P, -) : V -+ Mod, with R = Hom,(P, P ) is defined as Hom,(P, -) : V -+ Ab, only that the abelian groups Hom,(P, A ) have the structure of an R-right-module owing to the composition of morphisms of Homr(P, P ) and of Homw(P,A). A morphisni f : A -+ B then defines an R-homomorphism Hom,(P, A ) + Hom,(P, B). T h e functor Hom,(P, -) defines an isomorphism Homr(P, P ) g Hom,(Homr(P, P), HomV(P,P))

First, Hom,(P, -)

is faithful because P is a generator. Now let

f : Hom,(P, P) + Hom,(P, P ) be an R-homomorphism and let g = f(1p ) , then f ( r ) = f(1 r ) = f(1p ) r = g r = Hom,(P, g ) ( r ) , that

-

is, in this case Hom,(P, -) is surjective. Since P is finite projective, we get for families {Pi}iGIand {Pj}j,J with Pi P cx Pj

iGI

~ G J

n 1l iEI

jsJ

HornR@, , Rj)

Homr(Pi, pj) isI jeJ

where R, = Homg(P, Pi) g R, R j g R and the isomorphism is induced by Hom,(P, -). Hence, the functor Hom,(P, -) induces an equivalence between the full subcategory of the coproducts of copies of P in V and the full subcategory of coproducts of copies of R in Mod, (Section 2.1, Proposition 3). For each A E V there exists an epimorphism UieIPi + A. Thus we can construct for each A E V an exact sequence

212

4.

ABELIAN CATEGORIES

and correspondingly for each B E Mod, an exact sequence

where the index sets {i} = I and { j } = J certainly depend on A and B respectively. A and B are uniquely determined up to isomorphisms by f and g respectively as cokernels of these morphisms. If we apply to the first exact sequence the functor Hom,(P, -), then we get an exact sequence of the form of the second exact sequence because P is projective and thus Hom,(P, -) is exact. Then g has the form Hom,(P, f). T o each B there exists a g = Homw(P,f).Thus B = Hom,(P, Cok( f )). Each morphism c : A + A’ in %? induces a commutative diagram with exact sequences

-1l - 4- -

upi

f

f’

Pj*

Pi

b l

Pi#

A

0

c1

A‘ +0

since the coproducts of copies of P are projective. Correspondingly, we get for each R-homomorphism x : B -+ B’ a commutative diagram with exact sequences

IJR~8c

R~-+ B -0

The pair (x, y ) has the form (Hom,(P, a), Hom,(P, b)). Furthermore, determined by ( a , b ) and similarly z is uniquely determined by ( x , y ) as morphisms between cokernels. Thus z = Hom,(P, c), that is, Hom,(P, -) is full. Since P is a generator, Hom,(P, -) is also faithful and thus an isomorphism on all morphism sets. Thus the hypothesis for Section 2.1, Proposition 3 are satisfied and Hom,(P, -) is an equivalence of categories. Let 9: %? -+Mod, be an equivalence of categories and 9 : Mod, -+ % be the corresponding inverse equivalence. Then is left adjoint to 9, so Hom,(SR, -) g Hom,(R, 9-) 9 as functors. Furthermore, R Hom,(R, R) Hom,(gR, 9R).Since R is a progenerator in Mod,, also 9 R is a progenerator in W. This proves the theorem. c is uniquely

T h e categorical properties of module categories are also satisfied by cocomplete abelian categories with a progenerator by this theorem. In particular, we have the following corollary.

4.1 1

213

MODULE CATEGORIES

COROLLARY 1. A cocomplete abelian category with a progenerator is a Grothendieck category and has an injective cogenerator. Let R, S , and T be rings and ,AS,S B T , and R C T be bimodules. If we denote the R-S-bimodule homomorphisms by Horn,-,(-, -), then it is easy to verify that the isomorphism which defines the adjointness between the tensor product and the Hom functor preserves also the corresponding operator rings such that we get a natural isomorphism for the bimodules A, B, and C

0S B T

9

RCT)

HomS-T(SBT

*

SHomR(RAS

9

RCT)T)

where we gave the operator rings in each case explicitly. For E Hom,(,A, , ,CT), a E A, s E S, and t E T we define ($)(a) = (f(as))t so that Hom,(,A, , R C T is ) an S-T-bimodule. f

THEOREM 2 (Morita). Let rings R and S and an R-S-bimodule ,PS be given. Then the following assertions are equivalent: (a) The functor P 0, - : ,Mod -+ ,Mod is an equivalence. (b) The functor - 0, P : Mod, + Mod, is an equivalence. (c) The functor Hom,(P, -) : ,Mod -+ ,Mod is an equivalence. (d) The functor Hom,(P, -) : Mod, -+ Mod, is an equivalence. (e) ,P is a progenerator and the multzplication of S on P defines an Hom,(P, P)O. isomorphism S (f) P, is a progenerator and the multiplication of R on P defines an isomorphism R Hom,(P, P).

Proof. T h e equivalence of (d) and (f) was proved in Theorem 1. T h e equivalence of (c) and (e) follows by symmetry if we observe that by our definition endomorphism rings operate always on the left side whereas S operates on P from the right side. T h e equivalence of (a) and ( c ) and of (b) and (d) can be obtained - and - ORP are left adjoint to the because the functors P (9, functors Hom,(P, -) and Hom,(P, -) respectively. T o show the equivalence of (e) and (f) we need some prerequisites. T h e bimodule ,PS is a generator in Mod, if and only if there is a bimodule ,QR and an epimorphism Q 8, P + S of S-S-bimodules. In fact, let P be a generator and Q = Hom,(P, S ) and the evaluation as homomorphism. If Q 0, P + S is an epimorphism, then there exists an epimorphism UgEQ Pq + S with Pq = P . Since S is a generator, P is also a generator. Lct , P S , ,QR = Hom,(P, S ) , and R = Hom,(P, P ) be given. Then there exists an R-R-homomorphism q~ : P 0, Q R which is defined --f

214

4.

ABELIAN CATEGORIES

by rp(p @ q ) ( p ’ ) = p q ( p ’ ) where q ( p ’ ) E S. P, is finitely generated and projective if and only if rp is an epimorphism. In fact, if P is finitely generated and projective and if { p , ,..., p , } generates P, then there exists S. @ e,S -+ P with ei b p i and eiS an epimorphism g : e,S @ Since P is projective there exists a section f : P -+ e,S @ * - * @ e,S. This induces homomorphisms f i : P-, S. Then

P

=g

m )

=

c

g(eift(P)) =

c

Pifi(P)

for all p E P, that is, rp(xpi @ qi) = 1,. Since rp is an R-R-homomorphism, rp is an epimorphism. Conversely, if rp is an epimorphism, then there exist finite families { p i } and { f i } with p = x p i f i ( p ) for all p E P. Let {ei} be a finite family of elements with the same index set, then we define P -+ e,S @ @ e n s by p I+ e i f i ( p ) and e,S @ ... @ e,,S + P by ei b p i . Then

x

(P+e,S@...@e,S+P)

hence P is finitely generated. Since e,S @ P is projective.

=

1,

.-.@ e n s is projective,

also

Assume that ( f )holds. Then we have an epimorphism P @,Q -+R. Hence ,P is a generator. Furthermore, this epimorphism induces a homomorphism Q + Hom,(P, R) of S-R-bimodules. Since Q @, P -+ S is an epimorphism, I E S occurs in the image of this homomorphism. So 1 E Hom,(P,P) occurs in the image of Hom,(P,R) 0, P -+ Hom,(P,P). This Hom,(P, P)-Hom,(P, P)-homomorphism is an epimorphism. Hence, ,P is finitely generated and projective, so it is a progenerator. We still have to show that S Hom,(P, P)O by the homomorphism induced by the right multiplication. Let p s = 0 for all p E P, then s = Is = f i ( p , ) s = f i ( p i s ) 0. If f E Hom,(P, P), then f ( p ) = f ( P 1 , ) = f ( X P f i ( P i ) ) = f ( E F ( P Ofi)(Pi))= X d P @ f i > f ( P i ) = p ( x f i ( f ( p i ) ) ) .So (e) is satisfied. By symmetry, one shows that (e) implies (f)

,

x

1

We call P an R-S-progenerator if P satisfies one of the equivalent conditions of Theorem 2.

LEMMA 2. Let 9 and B be additive functors from ,Mod to sMod. Let q : g -+ B be a natural transformation. If q ( R ) : S ( R ) ---t B(R) is an isomorphism then q ( P ) : S ( P ) -+ 3 ( P ) is an isomorphism for all finitely generated projective R-modules P. Proof. Let P @ P ’ E R @ @ R = Rn. Since 9and B are additive we have that q(Rn) : 9 ( R n ) ---f Y(Rn) is an isomorphism for S ( R n )

4. I 1

215

MODULE CATEGORIES

(F(R))"and g(R")ili (Y(R))". T h e injection P + R" and the projection Rn -+ P induce a commutative diagram

- 1-1 -1

S(P)

F(R")

F(P)

9(P)

%(I?")

9(P)

where the middle morphism is an isomorphism. T h e left square implies that q(P) is a monomorphism, the right square that 7 ( P )is an epimorphism. and 9 are bifunctors and if we This lemma certainly still holds if 9restrict our considerations to one of the arguments. Two applications of this lemma are the natural transformation A

0RB 3 a @ b I+

( f H f ( a )b) E HOmR(HOmR(A, R ) , B )

which is natural in A and B and the natural transformation HOmR(A, R)0RB 3 f @ b H ( a H f ( U ) b) E HOmR(A, B )

which is also natural in A and B. For these natural transformations, we HomR(HomR(R,R), B ) and Hom,(R, R) 0, B have R ORB HomR(R,B). In particular we get for an R-S-progenerator RPs isomorphisms between the following functors: P

Os- g Hom,(Hom,(P,

S), -)

HOmR(HOmR(P, R), -) P Homs(P, -) E - asHom,(P, S ) - @R

HOmR(P, -)

HOmR(P, R ) OR-

COROLLARY 2. Let P be an R-S-progenerator and let Q Then

=

HomR(P,R).

(a) Q is an S-R-progenerator. (b) Q Hom,(P, S) as S-A-bimodules. (c) Center(R) Center(S). The lattice !I!(RP) of R-submodules of P is isomorphic to the lattice (d) 21(sS)of left ideals of S. Correspondingly, we have %(PS)

%(R&!), a(QR)

21(SS),

'lz(.SQ)

%(RR)

and

B(J's)

EZ %(sSs) E S ( 8 R )

z ~(SQR)

4.

216

ABELIAN CATEGORIES

Proof. (a) By Lemma 2, Hom,(Q, -) : Mod, -+Mods is an equivalence of categories. (b) P 8, - is adjoint to Hom,(P, -). Thus by the preceeding remark Hom,(Hom,(P, S ) , -) is adjoint to Q OR-. But also Hom,(Q, -) is adjoint to Q OR-. Hence, Q Hom,(P, S) as S-modules. Since Ro is the endomorphism ring of sQ as well as of ,Hom,(P, S ) , the isomorphism is an S-R-isomorphism. (c) We show that between the elements of the center 3 ( R ) and the endomorphisms of the identity functor 9 of .Mod there is a bijection which preserves the addition of natural transformations and of elements of 3(R)as well as the composition of natural transformations and the multiplication in 3(R). Since between the endomorphisms of the identity functor of .Mod and the endomorphisms of the identity functor of ,Mod there exists a bijection which preserves all compositions, this proves (c). Let p : 9 --t 9 be an endomorphism of the identity functor of .Mod. p determines an R-homomorphism p(R) : R + R . Let r p = p(R)(l), then p(R)(r)= rp(R)(l) = rr, , For each R-module A and each Rhomomorphism f : R .--f A we get a commutative diagram

Hence,fp(R)(l) = p ( A ) f (l), that is, for all A E A , we havep(A)(a) = r,u because f can always be chosen such thatf(1) = a (R is a generator). For all r E R, we have YY, = p(R)(r)= rpr, hence r , E 3(R). Now let p1 9 pz : 9 9 be given. Then (P1 p,)(R)(l) = (Pl(R) Pz(R))(l) = Pl(R)( 1)+P Z W ( 11 and (PlPZ)(R)(1)=PdR)Pz(R)( 1)=(Pz(R)(1>)(PdR)( 1)I. Conversely, the multiplication with an element of the center defines an R-endomorphism for each R-module A . These R-endomorphisms are compatible with all R-homomorphisms, and hence define an endomorphism of 9.This application is inverse to the above given application. (d) T h e equivalence Hom,(P, -) preserves lattices of subobjects. Hom,(P, P)O g S implies the first assertion. Multiplication with elements of S defines R-homomorphisms of P. These are preserved by Hom,(P, -) as multiplications because for s, s' E S considered as elements of S as well as right multiplicators of P we get Hom,(P, s)(s') = s . s' = (s's) by S Hom,(P, P)O. T h e given isomorphism of lattices carries R-S-submodules of P over into S-S-submodules of S. Conversely, -+

+

+

4.12

217

SEMISIMPLE AND SIMPLE RINGS

the inverse equivalence carries S-S-submodules of S over into R-Ssubmodules of P because we also have Hom,(S, S)O S. T h e other lattice isomorphisms follow by symmetry. We also observe that Hom,(Q, R) E Hom,(Q, S) g P as R-Sbimodules because of the remarks which follow Lemma 2. By the same R as R-R-bimodules and Q ORP reasons, we get P @,Q S as S-S-bimodules.

4.12 Semisimple and Simple Rings Among many other applications of the Morita theorems (Frobenius extensions, Azumaya algebras), the structure theory of semisimple and simple rings is one of the best-known applications of this theory. We want to present it as far as it is interesting from the point of view of categories. Let R # 0 be a ring. R is called artinian, if R is artinian as an object in .Mod. A left ideal (= R-submodule in .Mod) is called nilpotent if An = 0 for some n 3 1. A ring R is called semisimple if R is artinian and has no nonzero nilpotent left ideals. A ring R is called simple if R is artinian and has no two-sided ideal (= R-R-submodule) different from zero and R.

LEMMA 1. Each simple ring is semisimple. Proof. Let A # 0 be a nilpotent ideal in a simple ring R. An = 0 is equivalent to the assertion that for each sequence a , ,..., a, of elements of A we get a , a, = 0. We show that C = A for all nilpotent ideals A is a two-sided ideal. It is sufficient to show that for each a E A and r E R the element ar is in a nilpotent ideal. We have ar E Rar and (r,ar) .*.(r,ar) = (r1a)(rr2u) (rr,a) r = Or

=

0

hence (Rat-), = 0. A # 0 implies C # 0. Since R is simple, R = C, hence 1 E C. Thus 1 E A, *-* A, for certain nilpotent ideals. T h e sum of two nilpotent ideals A and B is again nilpotent. In fact let An = Bn = 0, then (a,b,) (b,-,u,) b, = 0. Thus (anb,) = a,(b,a,) A B is nilpotent. This proves that 1 E R is an element of a nilpotent ideal, hence 1" = 0. This contradiction arose from the assumption that R has a nonzero nilpotent ideal. Consequently, R is semisimple.

+ +

+

LEMMA 2. If R is a semisimple ring, then each ideal of R is a direct summand.

4.

218

ABELIAN CATEGORIES

Proof. Since R is artinian, there exists in the set of ideals which are not direct summands a minimal element A (in case that this set is not empty). If A contains a proper subideal B C A, then B is a direct summand of R, hence, there is a morphism R -+ B such that ( B -+ A 3 R -+ B) = Is. Thus B is also a direct summand in A and we have A = B @ C. But also C is a direct summand of R. T h e morphisms R + B and R -+ C induce a morphism R B 0C such that(A = B @ C + R - + B @ C ) = I . . I f A i s n o t a d i r e c t s u m m a n d in R, then A must be a simple ideal. For some a E A , we have A a # 0 because otherwise A2 = 0. Since A is simple we have Aa = A hence ( A -+ R 5 A ) g 1, . Therefore, the set of ideals which are not direct summands of R is empty. --f

LEMMA 3. Let R be a semisimple ring, then all R-modules are injective and projective. Proof. We apply Section 4.9, Corollary 2 to the generator R. Since each ideal A is a direct summand of R, for each R-module B, the group Hom,(A, B ) is a direct summand of Hom,(R, B ) ; hence the map Hom,(R, B ) -+ Hom,(A, B) is surjective. T h u s all objects are injective. For all exact sequences 0 -+ A -+ B -+ C -+0, the morphism A -+ B is a section. Hence each epimorphism B -+ C must be a retraction. By Section 4.9, Lemma 2, each R-module is projective.

LEMMA 4. Each finite product (in the category of rings) of semisimple rings is semisimple. Proof. It is sufficient to prove the lemma for two semisimple rings R, and R, . Let R = R, x R, . If we recall the construction of the product of rings in Section 1.1 1 and the theorem of Section 3.2, then it is clear that R, and R, annihilate each other and that R = R, 0R, as R-modules. Let p : R -+ R, be the projection of the direct sum onto R, . Let A i be a descending sequence of ideals in R. Then p ( A i ) is a descending sequence of ideals in R, . Let K i= Ker(A, -+p(A,)). T h e Ki form a descending sequence of ideals in R, . T h e last two sequences become constant for i >, n. T h u s we get a commutative diagram with exact sequences

1 O-Kn-

An-p(An)-O

4.12 where A,+j

SEMISIMPLE AND SIMPLE RINGS

21 9

C A,,

This morphism is also an epimorphism. In fact let E A,+j with p(a,+J = p(a,). Hence, a, - a,.+i E K, = Kn+ C A,+i . Thus also, a, E Anti . Therefore R is artinian. Let A C R be a nilpotent ideal with An = 0, then for a E A also Ra, = R,a, R,a, with a{ E Ri . (Ra)" = 0. We have Ra = Ra, In fact

a, E A,, , then there exists an a,+i

+

Tlal

Hence, (R,a,

+

+

y2a2

=

R2ap)n =

(I1

+

+

T2)(Ql

(R,aJn

+4

+ (R2a2)n

=0

and consequently a, = a, = a = 0, since R, and R, have no nonzero nilpotent ideals. Therefore R is semisimple.

THEOREM 1. If R is semisimple, then R

=

A, @

0A , ,

where the

Ai are simple left ideals in R. Proof. Since each R-module is injective, each coproduct of injective modules is injective. By Section 4.10, Corollary 3 R is noetherian. Each indecomposable injective object is simple because all objects are injective. By Section 4.10, Theorem 4, R may be decomposed into a coproduct of simple left ideals. Since R is finitely generated, Section 4.10, Lemma 3 holds, that is, R may be decomposed into a finite direct sum of simple left ideals.

THEOREM 2. The ring R is simple if and only matrix ring with coeficients in a skew-field.

if R

is isomorphic to a full

Proof. A skew-field is a not necessarily commutative field. A full matrix ring over a skew-field is the ring of all n x n matrices with coefficients in the skew-field. I t is well known that such a ring is isomorphic to the endomorphism ring of an n-dimensional vector space over the skew-field K. A vector space of finite dimension is a progenerator. If we denote the full matrix ring by M,(K), then the categories of Kmodules (K-vector spaces) and of M,(K)-modules are equivalent. Since the n-dimensional K-vector space K" is artinian, also M,(K) is artinian by Section 4.11, Corollary 2. Since K has no ideals, also M,(K) has no two-sided ideals by the same corollary. Hence M,(K) is simple. Let R be simple and P be a simple R-module, then P is finitely generated and projective by Lemma I and Lemma 3. Let K = End,(P). Then K is a skew-field. In fact, let f : P P be a nonzero endomorphism of P, then the image off is a submodule of P,hence coincides with P ---f

220

4.

ABELIAN CATEGORIES

since P is simple. Also the kernel off is zero, hence f is an isomorphism and has an inverse isomorphism in K . This assertion, which holds for all simple objects in an abelian category, is called Schur’s lemma. The evaluation homomorphism P OKHom,(P, R) --t R is an R-Rhomomorphism. The image of this homomorphism is a two-sided ideal in R. Since P is simple, there exists an epimorphism R + P. Since P is projective, this epimorphism is a retraction and there is a nonzero homomorphism P -+ R. Therefore, the image of the evaluation homomorphism is nonzero. Since R is simple, the image must coincide with R. The evaluation homomorphism is an epimorphism. In the proof of Section 4.1 1, Theorem 2 we observed that this condition is sufficient for the fact that P is a generator. Hence P is an R-K-progenerator. By Section 4.11, Theorem 2(f), R g Hom,(P, P) and P is a finitely K generated projective K-module, that is, a finite dimensional K-vector space.

THEOREM 3. For the ring R the following assertions are equivalent: (a) R is semisimple. (b) Each R-module is projective. (c) R is a finite product (in the category of rings) of simple rings.

Proof. Lemma 3 shows that (a) implies (b). Lemma 4 shows that (c) implies (a). Thus we have to show that (b) implies (c). Since each R-module is projective, each epimorphism is a retraction. Then each monomorphism is a section as a kernel of an epimorphism. This means that, by Section 4.9, Lemma 2, each R-module is an injective R-module. Each R-module may be decomposed into a coproduct of simple R-modules as we saw in the proof of Theorem 1 . There are only finitely many nonisomorphic simple R-modules A, . In fact if A, is simple, then there is an epimorphism R + A, which is a retraction. Hence A, is a direct summand of R up to an isomorphism. By Section 4.10, Theorem 4, A, occurs up to an isomorphism in a decomposition of R into a coproduct of simple R-modules. By Section 4.10, Theorem 5, Ai is isomorphic to a direct summand of R in the decomposition given in Theorem 1 . Let El ,..., E , be all classes of isomorphic simple R-modules. Let @ A, with simple R-modules A, be given. We collect the R = A, @ isomorphic At of this decomposition, which are in E l , to a direct sum Ail @ *.- @ A,* = B, , Correspondingly we collect the A , in Ei to @ B, . Since there are only a direct sum Bj . So we get R = B, @ zero morphisms into nonisomorphic simple R-modules, and since all simple R-modules in Bi are isomorphic because of the uniqueness of the

4. I3

22 1

FUNCTOR CATEGORIES

decomposition, there exists only the zero morphism for different i and j between Bi and B, . For bj E B, the right multiplication bj : Bi + Bj is an R-(left)-homomorphism. This proves that BiBi = 0 for i # j and B,Bi C Bi. Each Biis a two-sided ideal, and the Bi annihilate each other. In the decomposition R = B, @ * - . @ B, we have 1 = e, *-e, . For bi E Bi we have bi = 1bi = eibi . Hence ei operates in Bi as a unit, that is, Bi is a ring and R the product of the rings B, ,..., B,. Each B,-module is an R-module if one has the Bj with j # i as zero multipliers for the B,-modules. T h e R-homomorphisms and the Bi-homomorphisms between the Bi-modules coincide. Hence all B,-modules are projective. By construction, Bi is a direct sum of simple isomorphic R-modules, which are simple and isomorphic also as B,-modules. Let P be such a simple Bi-module, then P is finitely generated and projective and also a generator, since Bi = P @ @ P. Hence P is a Bi-K-progenerator with a skew-field K , where we used Schur’s lemma. As in Theorem 2 we now have Bi End,(Km), that is, a simple ring.

+ +

We conclude with a remark about the properties of simple rings which may now be proved easily.

COROLLARY 1. The center of a full matrix ring over a skew-field K is isomorphic to the center of K . Proof. T h e category of modules over a full matrix ring over K is equivalent to the category of K-vector spaces. By Section 4.11, Theorem 2 and Section 4.1 1 , Corollary 2(c) the assertion is proved.

COROLLARY 2. Let R be a simple ring. Then each finitely generated R-module P is a progenerator and Hom,(P, P ) is a simple ring. Proof. T h e category of R-modules is equivalent to the category of K-vector spaces with a skew-field K. I n .Mod the assertion is trivial.

4.13

Functor Categories

T h e results of this section shall mainly prepare the proof of the embedding theorems for abelian categories presented Section 4.14. Therefore we shall restrict ourselves to the most important properties of the functor categories under consideration. Let d and V be abelian categories and let the category d be small. By Section 4.7, Proposition 1 , we know that F u n c t ( d , 9)is an abelian

222

4. ABELIAN

CATEGORIES

category. We form the full subcategory % ( dV) , of Funct(cc4, V) which consists of the additive functors from d to V.

PROPOSITION. 21(d, '3) is an abelian category. Proof. We know that limits preserve difference kernels and that colimits preserve difference cokernels (Section 2.7, Corollary 2). By Section 4.6, Proposition 1, limits and colimits are additive functors. Since by Section 2.7, Theorem 1 , limits and colimits of functors are formed argumentwise, a limit as well as a colimit of additive functors in F u n c t ( d , V) is again an additive functor. T h u s the full subcategory '$I(&, V) of F u n c t ( d , V) is closed with respect to forming limits and colimits. In %(d, V) there exist kernels, cokernels, finite direct sums, and a zero object and they coincide with the corresponding limits and colimits in F u n c t ( d , V). Furthermore, each isomorphism in V) is also an isomorphism in %(A?',V) F u n c t ( d , V) which is in %(d, because 2 l ( d , V) is full. Therefore, '$I(&, V) is an abelian category. For our considerations we need still another full subcategory of F u n c t ( d , V), namely 2 ( d , V), the category of left-exact functors from d to U. Obviously Q ( dV), is also a full subcategory of %(d, U) because each left-adjoint functor is additive. We want to investigate 2 ( d ,V) further and we want to show in particular that this category is abelian. I t will turn out that the cokernels formed in 2 ( d ,U ) are different g).This means that the embedding from the cokernels formed in %(d, functor is not exact. T o construct the cokernel in Q(cc4,V) we shall show Lhat 2 ( d , $2) is a reflexive subcategory of '$IV). (& For, this purpose, we solve the corresponding universal problem with the following construction. Let A E d.Denote the set of monomorphisms a : A -+ X in cc4 with domain A and arbitrary range X E cc4 by S(A). Observe that d is small. T o S ( A ) we construct a small directed category T ( A )with the elements of S ( A ) as object. We define a < b, that is, there is a morphism from a to b in T ( A )if and only if there is a commutative diagram X

Y

4.13

223

FUNCTOR CATEGORIES

in d, that is, if b may be factored through a. This factorization x need not be uniquely determined. On the other hand, by definition of the directed category, there can exist at most one morphism between a and b in T ( A ) .We call x the representative of this morphism. Trivially a a is satisfied by the identity. Also the composition of morphisms in T ( A ) holds because morphisms may be composed in a?.Given objects a and b in T ( A ) ,we get a c in T ( A )with a c and b c by the following cofiber product




is commutative. T h e morphism S ( A )-+ ( R 9 ) ( A )is a natural transformation, which will be denoted by p : F--t (RF).

4.

228

ABELIAN CATEGORIES

LEMMA 2. Let '3 : a2 + V be a left exact functor and pl : 9 3 '3 be a natural transformation. Then there exists exactly one natural transformation $ : (R9)--+ '3 such that $p = pl. Proof. Let a E T(A). Then, be the left exactness of g,we get a commutative diagram with exact rows +

0

-

+S(X)

Y(A)

-

P(X)

P(Cok a)

-

Y(Cok u )

where .FA*(a)+ S ( A ) is uniquely determined by cp. If a this uniqueness the diagram

< a', then by

is commutative. Hence we can factor pl(A) through ( R 9 ) ( A )= li$ .FA*:

where $ ( A ) is uniquely determined by this property, for ( R F ) ( A )is a direct limit. We still have to show that $ is a natural transformation. Letf : A -+ B be a morphism in A. Let b = T ( f ) ( a ) .Then by two-fold application of the first diagram in this proof, together with the construction of b, we get that

PA*(a)

-

'%*(b)

4.13

FUNCTOR CATEGORIES

229

is commutative. T h e direct limit preserves this commutativity, that is, is a natural transformation.

t,h

LEMMA 3. I f f : A monomorphism in %.

-+

B is a monomorphism in d,then (R.F)(f ) is a

Proof. Similar to the definition of Sf* we define a natural transformation Sf+ : FA*T+(f ) -+ SB* by the commutative diagram

- 9( Y)

0 4 SA *(bf)

0

-

I

9 ( C o k bf)

.FB*(b)--+F( Y )---

9(Cok b)

As for Ff*, here again one proves that Sf+ is a natural transformation. But the above diagram implies also that Ff+(b) : F'*(bf) -+ S B * ( b ) is a monomorphism because SA*(bf) -+ .F(Y) is a monomorphism. Since, by hypothesis on %, the Grothendieck condition holds, also lim Ff+ : lim P A * T + ( f+ ) lim SB* +

--+

-+

is a monomorphism (Section 4.7, Theorem 1). Let a E T ( A ) and b = T ( f ) ( a ) .T h e commutative diagram

b

B--+Y

implies that .Ff*(a)may be factored through

*+(T(f)(4) : % * T + ( f ) T ( f ) ( a > % * T ( f ) ( a ) +

where the morphism S A * ( a )+ SA*T+(f ) T(f )(a) is induced by a T+(f ) T ( f )(a). This factorization is preserved by the direct limit. Observe that the morphisms .FA*(.)-+ F A * T + ( f ) T(f )(u) give the identity after the application of the direct limit. This implies the assertion of the lemma.



( Y X ) E X ) A (A x, Y , Z)((XY) E A (YZ) E

x

=+ ( X X ) E

..-- Rel(X)

MaP(X) F map on X FmapfromXto Y bijective (F)

A

Map(F)

:i=

A

E

X)

x

XI,

Un(X), 9(F)

=

X,

:= F m a p o n X A %(F)C Y , :.r

Un(F)

A

Un(F-l).

Let F be a class and x be a set. Then F(x) is uniquely defined by (((V!Y)((YX)E F ) )

((%)

x>

EF))

*

(V!Y)((YX)

((1

* F(4 =

Let F map from X to Y be given. F is called injective, if Un(F-l). F is called surjective if %(F) = Y . Instead of F map from X to Y , we often write F : X + Y or X 3 x I-+ F(x) E Y or X Y. Observe that the arrow F-+ is used between sets which are assigned to each other, whereas the arrow --+ is used between sets or classes the elements of which are assigned to each other. A family F of elements of Y with index set X is ( F map from X to Y ) . We have the following rules of set theory: (a) (b) (C) (d) (e) (f)

00 = U ;

nlr= 0;

u0

CXCU; X = Y o X C Y A YCX; 0

B(U) = U; %(U) = U; cpW) = U;

S(@;

C(U);

= 0;

uU=U;

254 (g) (h) (i) (j)

(k) (1) (m) (n) (0)

-

APPENDIX

S(X n Y ) A S('P(X))A S(U X ) ; 3 s(nX I ; S ( X ) A ( Y C X )* S ( Y ) ; S(X) A S ( Y ) =. S(X x Y ) A S(X u Y); F map on x * S(F) A S(%(F)) A S(F((x))); C ( X ) 3 C('P(X))A C(U X)A C ( X u Y ) A C ( X - y ) ; C ( X ) A Y # 0 * C(X x U); bijective(F) A X C B(F) A C ( X ) * C ( F ( ( X ) ) ) ; F map on A A G map on A * ((A u)(u E A F(u) G(u)) F = G). S(X)

x#

0

-

j.

Proof. It is trivial to verify (a), (b), (c), (d), and (e). (i) Let A = {(zz) I z E Y}, then A has uniquely defined values (Un(A)). By axiom (C4), we get y = Y, that is, S( Y). X) by (g) S(X n Y) trivially by (i). (i) implies also S(?(X)) and axioms (C2) and (C3). (j) {X, Y } is a set by axiom (A4). (g) implies S(X U Y). X x Y C '$'p(X u Y) implies S(X x Y) by (i) and (g). (k) S(%(F)) and S(F((x)))hold by axiom (C4). F C x x %(F) implies S(F). (l), (m), and (n) are proved analogously. (0) holds by definition of F(x). (f) er C X and the existence of a set (axiom(C1)) imply S ( 0 ) . Assume S(U). Then U E U and U E {U} contradicting axiom D. Hence CW. (h) y E X and X C y imply S( X).

S(u

n

n

T h e strong axiom of choice of Godel is equivalent to the axiom of choice we use here and is particularly suitable for the application in categories. (The equivalence of these two axioms holds only if the axiom of foundation holds.) T h e axiom of choice is Equ. Re1 R

a

( V X)(A u)(u E B(R) 3 (V! v)(v E X

A

(uv)

E

R))

To each equivalence relation R on B(R) there exists a complete system of representatives X n 'D(R). The axiom of choice is equivalent to the following axiom of THEOREM. choice of Godel ( V A)(Un(A)

A

(A x ) ( 1 ErnPtY(4. a ( V Y ) ( Y E

A

(YX) E4

(There is a class with uniquely dejined values (an application) which assigns to each nonempty set x one of its elements.)

255

FUNDAMENTALS OF SET THEORY

Proof. Assume that the axiom of choice holds. Let E be the class of the erelation: E = { ( x y ) I x E Y } . Let R

= {(WXYZ)

1

(WX) E

E

A

(YZ) E E

A

x = Z}

Then R is an equivalence relation on E. Let A be a complete system of representatives for R. If y # a , then there is exactly one x with ( x y ) E A C E. Thus A is the choice function for the strong axiom of choice of Godel. We shall only indicate the converse of the proof. Godel’s strong axiom of choice implies that U may be well-ordered. If R is an equivalence relation, then X

= {.x

1 x E B(R) A (A y)(y E B(R) A ( x y ) E R

=>

x