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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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.

<

3. S*-Category of pointed sets: A pointed set is a pair (A, a) where A is a nonempty set and a E A. A pointed map f from (A, a) to (B, b) is a map f : A + B with f (a) = b. The pointed sets form the objects and the pointed maps, the morphisms of this category. 4. Gr-Category of groups: A group consists of a nonempty set A together with a composition A x A3(a,b)~ab~A

such that the following axioms hold: (1) a(bc) = (ab)c for all a, b, c E A (2) there is an e E A with ea = ae = a for all a E A (3) for each a E A there is an a-l E A with aa-l = a-’a = e

4

1.

PRELIMINARY NOTIONS

A group homomorphism f from a group A into a group B is a map from A into B withf(aa’) = f ( a ) f ( a ’ ) .The groups form the objects and the group homomorphism.., the morphisms of this category.

5 . Ab-Category of abelian groups: A group A is called abelian if ab = ba for all a, b E A. The abelian groups together with the group homomorphisms form the category A b . 6. Ri-Category of unitary, associative rings: A unitary, associative ring consists of an abelian group A (whose composition is usually written as (a, b) e a b) together with a further composition

+

A xA3(a,b)~ab~A

such that the following axioms hold:

+ +

(1) (a b)c = ac + bc for all a, b, c E A (2) a(b c) = ab ac for all a, b, c E A (3) (ab)c = a(bc) for all a, b, c E A (4) there is a 1 E A with la = a1 = a for all a E A

+

A unitary ring homomorphism f from a unitary, associative ring A into a unitary, associative ring B is a map from A to B with f(a

+ a‘)

= f ( a ) +f(a’),

f(aa’) = f(a)f(a‘),

and f ( 1 )

=

1

The unitary, associative rings together with the unitary ring homomorphisms form the category Ri.

7 . .Mod-Category of unitary R-modules (for a unitary, associative ring R): A unitary R-(left-)module is an abelian group A [whose composition is usually written as (a, b) e a

+ b] together with a composition

R x A3((r,a)+~a~A such that the following axioms hold:

+ +

+ +

(1) r(a a’) = ra ra‘ for all r E R, a, a’ E A (2) ( I r’)a = ra r’a for all I , I’ E R, a E A (3) (rr’)a = r(r’a) for all I , I’ E R, a E A (4) l a = a for all a E A

A homomorphismffrom a unitary R-module A into a unitary R-module B is a map from A into B withf(a + a‘) = f ( a ) + f ( a ’ ) andf(ra) = t f ( a ) . The unitary R-modules together with the homomorphisms of unitary R-modules form the category .Mod. If R is a field, then the R-modules are called vector spaces.

1.1

DEFINITION OF A CATEGORY

5

8 . Top-Category of topological sptzces: A topological space is a set A together with a subset 0, of the power set of A such that the following axioms hold: (1) if Bi E 0, (2) if B,EO,

(3) (4) A

for i E I, then Uie,Bi E 0, for i = 1,..., n, then fiblB i ~ O ,

EOA ' A

The elements of 0, are called open sets of A. A continuous map f from a topological space A into a topological space B is a map from A into B with f - l ( C ) E O , for all C E O ~The . topological spaces together with the continuous maps form the category T o p .

9. Htp-Category of topological spaces modulo homotopy: Two continuous maps f and g from a topological space A to a topological space B are called homotopic if there is :i continuous map h : I x A + B with h(0, a ) = f ( a ) and h(1, a ) = ,?(a) for all a E A, where I is the interval [0, 11 of the real numbers. The open sets O,,, of I x A are arbitrary unions of sets of the form J x C, where J C I is an open interval and C E 8, . Homotopy is an equivalence relation for continuous maps. The equivalence classes are called homotopy classes of continuous maps. Juxtaposition of homotopic, continuous maps gives again homotopic, continuous maps. Thus the juxtaposition of maps defines a composition of homotopy classes which is independent of the choice of representatives. The topological spaces together with the homotopy classes of continuous maps and the just discussed composition form the category H t p .

10. Top*-Category of pointed topological spaces: A pointed topological space is a pair (A,a) where A is a nonempty topological space and a E A. A pointed continuous map f from ( A ,a ) to (B, b ) is a continuous map f : A -P B withf(a) = b. The pointed topological spaces together with the pointed continuous maps form th.e category T o p * .

I I . Htp*-Category of pointed tcpological spaces modulo homotopy: Two pointed continuous maps f and g from a pointed topological space (A, a ) into a pointed topological spa.ce (B, 6 ) are homotopic if they are homotopic as continuous maps and if h(r, a) = b for all r €1. T h e pointed topological spaces together with the homotopy classes of pointed continuous maps and the composition of homotopy classes as defined in Example 9 form the category Htp*.

6

1.

PRELIMINARY NOTIONS

12. Ordered set as a category: Let A be an ordered set in the sense of Example 2. A defines a category d with the elements of A as objects. For a , b E A = Ob d we define {(a, b)} if a < b Mord(a, b) = 0

otherwise

The transitivity of the order relation uniquely defines a composition of the morphisms. The reflexivity guarantees the existence of the identity. Since Mor,(a, 6 ) has at most one element, the composition is associative. 13. Group as a category: Let A be a group. A defines a category JZZ with exactly one object B and Mor,(B, B) = A such that the composition of the morphisms is the multiplication (composition) of the group. 14. Natural numbers as a category: The natural numbers form an ordered set with the order relation a < b if and only if a I b. As in Example 12, the natural numbers form a category.

15. Category of correspondences of sets: A correspondence from a set A to a set B is a subset of A x B. Iff C A x B and g C B x C let gf

= {(a,c) I a E A ,

cE

C , there is a b E B with (a, b) E f and (b, c) Eg}

The sets as objects and the correspondences of sets as morphisms form a category. 16. Equivalence relation as a category: Let M be a set and R be an equivalence relation on M , Let the objects be the elements of M. If ( m , m’) E R, then Mor(m, m f ) = {(m, m f ) } . If (m, m f )4 R, then Mor(m, m’)= 0 . This defines a category. A category V is called a discrete category if Morw(A, B ) = o for any two objects A # B in V and if Mory(A, A) = {IA} for each object A in %. Similarly to Example 12, every class defines a discrete category. Conversely, every discrete category may be interpreted as a class. Examples 12, 13, 14, and 16 are categories of a special type, namely those with only a set (instead of a class) of objects. If the objects of a category form a set, then the category is called a small category or diagram scheme. An explanation for the second name will be given in Section 1.8.

1.2 Functors and Natural Transformations We stressed already in Section 1.1 that, together with every kind of mathematical object, the corresponding maps have to be studied as well.

1.2

FUNCTORS AND NATlJRAL TRANSFORMATIONS

7

The mathematical objects we defined in Section 1.1 are the categories. The place of the maps will be taken by the functors. Let 9? and V be categories. Let 9consist of (1)

amap 0 b a 3 A t + F ( A ) ~ : O b V

(2) a family of maps {MordA, B ) 3f++F(f)E M o r u ( W 4 , ~ ( B ) ) )

for all A, B E Ob 93

9 is called a covariant functor if 9 complies with the following axioms:

(I) 9 ( I A ) = for all A E Ob 9 ( fg) = 9 ( f ) 9 ( g ) for all f E MOTa(& C), g E Morg(A, B ) (2) and for all A, B, C E Ob a Let 28 and ‘3be categories. Let 9 consist of

(I) a m a p O b a 3 A b F ( A ) E i O b V (2) a family of maps {Mor,e(4 B ) 3f.+ T(f> E Mor,(%(B), s ( A ) ) )

for all A, B E Ob 9Y ,F is called a contravariant functor* if 9 complies with the following axioms:

(1) 9 ( 1 J = lF(A) for all A E Ob a (2) S(fg) = 9 ( g ) 9 (f ) for all f E Mora(B, C),g E Mora(A, B ) and for all A, B, C E Ob 9? Since the (co- and contravariant) functors take the place of the maps, we shall often write 9: a --t ‘3if fF is a functor from the category L% to the category %. If there is no ambiguity, we shall also write .FA instead of 9 ( A ) and 9f instead of .F(f). A covariant functor will often be called only “functor.” If a, V, and 9 are categories and S : L% --+ V and 9 : V --+ 9 are functors, then let 3’9 : 9Y-+ 9 be the functor which results from the composition of the maps defining the functors 9and g.I n fact we have

gw,) = 9 ( 1 5 ( A ) ) =

1 m A )

and gs(fg)

=

S(*(f) sk))= 3%(f)Y S ( g )

8

1.

PRELIMINARY NOTIONS

Observe the change in the order of the morphisms if one of the two functors is contravariant. Thus 929 is covariant if both functors are simultaneously co- or contravariant. If one of the functors is covariant and the other one is contravariant, then 2 3 9 is contravariant. If X : 9 -+d is an additional functor, then the composition of functors is associative (X(23.F)= ( X 9 ) 9 )because of the associativity of the composition of maps. Idy : V + V denotes the functor with the identity maps as defining maps. Idy is a covariant functor. As above, we have for functors 9and B Idy9

=9

and

9Idy

=

9

After these considerations one should expect that the categories and functors themselves form a category (of categories). This, however, is not the case in the set theory we use. I n fact a category is in general no longer a set but a proper class. Thus, we cannot collect the categories in a new class of objects (see Appendix). In general, the functors, too, are proper classes and cannot be collected in morphism sets. But if we admit only small categories, every category, interpreted as a set of certain sets, is a set, and every functor is a set. Therefore, the category of small categories with functors as morphisms, Cat, may be formed. As an example, we want to mention only a special type of functor. Later on we shall study further examples of functors in more detail. All the categories S*, Gr, Ab, Ri, ,Mod, Top, and Top* have sets with an additional structure as objects. T h e morphisms are always maps compatible in a special way with the structure of the sets. T h e composition is always juxtaposition. If one assigns to every object the underlying set and to every morphism the underlying set map, then this defines a covariant functor into S,very often called a forgetfulfunctor. Instead of forgetting the structure on the sets completely, one can also forget only part of the structure. For example, the abelian groups are also groups, and the homomorphisms are the same in both cases. T h e rings are also abelian groups, and the ring homomorphisms are also group homomorphisms. So we get forgetful functors Ab + Gr and Ri +Ab respectively. Similarly, there are forgetful functors ,Mod +Ab and Top* + Top. If the topological spaces carry an additional structure (e.g., hausdorff, compact, discrete), respective categories are defined thereby. So we get forgetful functors into the category Top. The example Ab --t Gr and the aforementioned examples have an additional property. An abelian group is a group with a special property. Likewise, a hausdorff topological space is a topological space with a special property. The objects of one category are in each case also objects

1.2

FUNCTORS AND NATURAL TRANSFORMATIONS

9

of the other category. T h e morphisms of one category are morphisms of the other category. T h e composition is the same. T h e identities are preserved by the forgetful functor. A category d is called a subcategory of a category &? if Ob d C Ob B

Mord(A, B ) C Mors(A, B )

and

for all A, B E O b d , if the composition of morphisms in &' coincides and if the identity with the composition of the same morphisms in g, of an object in d is also the identity of the same object viewed as an object in a. Then there is a forgetful functor from &' to We note that Ri is not a subcategory of Ab. In fact, Ob Ri C Ob Ab is not true, although every ring can also be regarded as an abelian group. T h e corresponding abelian groups of two rings may coincide even if the rings do not coincide. T h e multiplication may be defined differently. Let 9 : --t % and % : &? --f %' be two covariant [contravariant] functors. A natural transformation cp : .% --+% is a family of morphisms {v(A): .%(A)+ %(A)}for all A E Ob 9such that we have v(B).%(f ) = %( f ) v ( A ) [v(A)g(f ) = 9(f ) p(B), respectively] for all morphisms f : A + B of &?. I n the following there will often be equalities between composed morphisms. T h e objects which are the domains and the ranges of the separate morphisms will not appear explicitly. Thus, these equations are difficult to comprehend. So we take a detailed representation using arrows, as we already took for single morphisms. This will be called a diagram. I n the case of a natural transformation between covariant functors, the defining equation

a.

dB1 F(f)= B(f)d 4 may be illustrated by the diagram

s ~ l I9f 9-A

FB

,?(A)

C(B)

BA

BB

T o follow the arrow v(A)from .%A to %A and then the arrow 9fto %B substitutes the arrow %fv(A) from P A to %B. Correspondingly, v(B)Sfruns through 9 B . T h e condition that these two morphisms coincide will be expressed by saying that the diagram is commutative. A diagram with arbitrarily many objects and arrows is called commutative if, for any two objects of the diagram, the morphism obtained by following

10

1.

PRELIMINARY NOTIONS

a path between the two objects in the direction of the arrows is independent of the choice of path. If there is no ambiguity, we shall often write pA instead of p(A). A natural transformation is often called a functorial morphism. If p : 9+ ’23 and $ : B + Z are natural transformations, then so is $p with +?(A) := +(A)p(A). We have (p+)p = p($p) because of the associativity of the composition of morphisms. T h e family {lPA : 9 A +9 A ) defines a natural transformation id,: 2F + 9. For all natural transformations p :F -+G and $ : G +F, we have idF$

=$

and

pids

=

Here again it seems as if the functors together with the natural transformations form a category. Here again, set theoretic difficulties arise from the fact that the functors are generally proper classes and cannot be collected in a class of objects. If d is a small category and 97 an arbitrary category, then d is a set and 33 a class. A functor F :&+A?, originally defined as a map, is a set by Axiom C4 (Appendix) if it is interpreted as a graph. Similarly, a natural transformation between two functors from d to 33, being a family of morphisms with an index set O b d,is a set. T h e natural transformations between two functors from d to 9are a set, being a subset of the power set of MorB(9A, ’23A) Therefore, the functors from a small category d into a category 9, together with the natural transformations, form a category F u n c t ( d , which we call the functor category. If the categories d and are not explicitly given, the functor is not considered only as a graph. One also asks that functors between distinct pairs of categories are distinct so that in this general case a functor may well be a proper class, even if the domain of the functor is a small category. If d is the empty category, then F u n c t ( d , 9) consists of exactly one functor and exactly one natural transformation, the identity transformation. An important example of a natural transformation will be presented in the following section.

uAEd

a),

1.3 Representable Functors Let ‘Xbe a category. Given A E ‘XandfE Mory(B, C), we define a map Morv(A,f): MorW(A,B)

by Mor,(A,f)(g)

--f

Mor,(A, C )

:= fg for all g E Mor,(A, B ) and a map

Mot-& A ) : Morw(C, A ) -+ Mor&

by Mor,(f, A)(h) := hf for all h E Mor,(C, A).

A)

1.3

REPRESENTABLE FUNCTORS

11

LEMMA,Let V be a category and A E V. Then Mor,(A, -): V -+ S with Ob %? 3 B F+ Mory(A, B) E Ob S

Mory(B, C) 3fw Mor,(A,f)

E

lVIors(Mor&4, B), Moryp(A, C))

is a covariant functor. Furthermore, :.War,(-, A) : V -+ S with O b g 3 B t t Morv(B, A) E Ob S Mory(B, C) 3fw Morv(f, A) E lMor,(Mory(C, A), Mory(B, A))

is a contravariant functor. Proof. We prove only the first assertion. T h e proof of the second assertion is analogous and may be trivially reduced to the first assertion by later results on the duality of categories. Mory(A, I,)(g) = 1,g = g implies Mor,(A, l B ) = lMor(A,B). Let f , g E V be given such that the domain off is the range of g . Then f g exists, For all morphisms h, we have Mor,(A,fg)(h)

= (fg)h = f(gJi) =

Morw(A,f) Morw(A,g)(h)

whenever these expressions are defined. T h e functors of this lemma are the most important functors in the theory of categories. So they get a special name: Mory(A, -) is called covariant and Mor,(-, A) contravariant representable functor. A is called the representing object. Now we want to give an example of a natural transformation. Let A and B be two sets. T h e map A x Mor,(A, B) 3 ( a , f ) t . f ( a )

EB

is called the evaluation map. For fixed a E A,the evaluation map defines a map from Mor(A, B) to B, the evaluation of each morphism f at the argument a . Thus we obtain a map A -+ Mor(Mor(A, B), B), labeled y ( A ) . Mor(-, B) is a contravariant functor from S to S. T h e n Mor(Mor(--, B), B), as a composition of two contravariant functors, is a covariant functor from S to S. Now we show that y is a natural transformation from Id, to Mor(Mor(-, B), B). Let g: A -+ C be an arbitrary map of sets. We have to check the commutativity of the diagram

4 A

cp(A)

Mor(Mor(A, B), B) ~Mor(Mor(g.B),B)

Mor(Mor(C, B), B)

1.

12

PRELIMINARY NOTIONS

For a E A both Mor(Mor(g, B), B)v(A)(a)= y(A)(a)Mor(g, B ) and y(C)g(a) are maps from Mor(C, B) into B. Let f~ Mor(C, B). Then v(A)(a) Mor(g, B)(f 1 = dA)(a)(fg)

= f g ( 4 = f(g(a>>= 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.

<

<

Proof. Let f g and g h be subobjects of B. Then f = gk and = hk’, hence f = hk’k, that is, f h. Furthermore, we get f f by f = f 1, if A is the domain off. Finally, iff g and g f, then f and g are equivalent, so f = g.

g

<

<

<

<

Instead of the monomorphism which is a subobject we shall often give only its domain and call the domain a subobject. Thus we can again interpret a subobject as an object in V, tacitly assuming that the corresponding monomorphism is known. Observe that a monomorphism is not uniquely determined by the specification of the domain and the range so that an object may be a subobject of another object in different ways. I n S,for example, there are two different monomorphisms from a one point set into a two point set. I f f g for subobjects f : A + C and g : B -+ C, then we often write A C B C C.

<

1.6

SUBOBJECTS AND QUOTIENT OBJECTS

21

T h e ordered class of the subobjects of an object B E %? is called the power class of B. If the power class of each object of a category V is a set, then V is called a locally small category. T h e n the power classes are also called power sets. Let V be a locally small category. Let U be a subset of the power set of the subobjects of B E V. A subobject A E U is said to be minimal in U if A’ E U and A‘ C A always implies A‘ = A . T h e power set of the subobjects of B E V is called artinian if, in each nonempty subset of the power set of the subobjects of B, there is a minimal subobject. A subobject A E U is said to be maximal in U if A’ E U and A C A’ always implies A’ = A . T h e power set of the subobjects of B E % is called noetherian if, in each nonempty subset of the power set of the subobjects of B, there is a maximal subobject. If the power set is artinian or noetherian, then we also call B an arthian or noetherian object respectively. If all objects of V are artinian or noetherian, then the category ‘3? is said to be artinian or noetherian respectively. A subset K of the power set of B is called a chain if for any two subobjects A, A’ E K we always have A C A’ or A’ C A. We say that B E %? complies with the minimum condition [maximum condition] for chains if each nonempty chain in the power set of B contains a minimal [maximal] element.

LEMMA 2. A n object B E V complies with the minimum condition [maximum condition] for chains ;f and only if B is artinian [noetherian].

Proof. If B is artinian, then in particular B complies with the minimum condition for chains. Let B comply with the minimum condition for chains and let U be a subset of the power set of B which does not contain a minimal subobject. Then to each subobject A, E U there is a subobject A,+l E U with Ai+l C A, and # A, . This will also be written as A,+l C A, , So we get a chain K with no minimal element in contradiction to the hypothesis. Thus B is artinian. T h e equivalence of the maximum condition for chains with the condition that B is noetherian may be shown analogously. One easily shows that the subobjects in S,Gr, Ab, or Ri are the subsets, subgroups, abelian subgroups or subrings with the same unit together with the natural inclusions. I n Top the subsets of a topological space equipped with a topology in such a way that the inclusion maps are continuous are the subobjects of the topological space. T h e socalled subspaces of a topological space have additional properties and will be discussed in Section 1.9. By dualizing we obtain the notion of the quotient object, the copower

22

1.

PRELIMINARY NOTIONS

class and the locally cosmall category. T h e discussed properties may be dualized similarly. The property of being a subobject is transitive in S, Gr, Ab, Ri, Top, S*, and Top*; that is, if A is a subobject of B and if B is a subobject of C, then A is a subobject of C. This, however, is not the case if one considers quotient objects, for example, in Ab, since the quotient object of a quotient object has as elements residue classes of residue classes whereas a quotient object has as elements residue classes (of the original object). So this transitivity cannot be expected in a general form and, in fact, is not implied by our definition of subobjects and quotient objects.

1.7 Zero Objects and Zero Morphisms An object A in a category V is called an initial object if Mor,(A, B ) consists of exactly one element for all B E V. T h e notion dual to initial object isfinal object. An object is called a zero object if it is an initial and a final object.

LEMMA1. All initial objects are isomorphic. Proof. Let A and B be initial objects. Then there is exactly one morphism f : A --t B and exactly one morphismg : B -+ A. T h e composition f g [ g f ] is the unique morphism l B [ I A ] which exists in Mor,(B, B) [Mor,(A, A ) ] .Thusf and g are isomorphisms.

LEMMA 2. A zero object 0 of a category V is in a unique way a subobject of each object B E V up to isomorphisms of zero objects. Proof. Since Morw(C, 0) consists of the unique morphism f : 0 + B is Mor,(C, f ) : Morw(C,0 ) -+ Mor,(C, object of B which represents f must morphic to 0.

exactly one element for all C E V, a monomorphism for all B, for B) is always injective. T h e subhave as domain a zero object iso-

A morphism f : A ---+ B in C is called a left zero morphism if f g = f h for all g , h E Mor,(C, A) and all C E '%?.Dually we define a right zero morphism. f is called a zero morphism iff is a right and left zero morphism.

LEMMA 3. (a) I f f is a right zero morphism and g is a left zero morphism and if f g is dejined, then f g is a zero morphism.

1.7

ZERO OBJECTS AND ZERO MORPHISMS

23

(b) Let A be an initial object. Then f : A + B is always a right zero morphism . (c) Let 0 be a zero object. Then f : 0 --t B and g : C --t 0 and consequently also f g : C -+ B are zero morphisms.

Proof. The assertions are direct con:sequences of the definitions of the particular notions.

A category V is called a category with zero morphisms if there is a family {O(A,B)E Morw(A, B) for all A, B E g} with fo(A.B)

= O(A.C)

and

O(B,C)g = O(A.B)

for all A, B, C E %? and all f E Mory(B, C) and g E Mor,(A, B). The O(A,B) are zero morphisms because f O(A,B)= O(A,c) = hO(,,B) , and correspondingly for the other side. 'The family {O(A,B)} of these zero morphisms is uniquely determined. For if {O:A,B)}is another family of zero morphisms, then O(A.B) = O(A,B)O;A.A)

= 0iA.B)

A,

for

Eg

LEMMA4. A category with Q zero object is a category with zero morphisms. Proof. The zero morphisms O(A,B) are constructed as in Lemma 3(c). The rest of the assertion is proved by the commutativity of the diagrams f

OI.4 B )

A--+B-C

0

A-

g

B oIB.CI -C

0

T h e category V is a category with zero morphisms if and only if the and sets Mor,(A, B) are pointed sets ;and the maps Mor,(f,-) Mor,( -, g) are pointed maps (in the sense of Section 1.1, Example 3). Thus %? is said to be apointed category. In %?the distinguished points of Mor,(A, B) are uniquely determined ;by the condition that Mor,( f, -) and Mor,( -, g ) are pointed set maps. In the category S an initial object is 0 and a final object is { a } . Zero objects do not exist. The only zero morphisms have the form 0 --t A. In the category S* each set with one point is a zero object. Thus there are zero morphisms between all objects,. Similarly, the set with one point

24

1.

PRELIMINARY NOTIONS

with the corresponding structure is a zero object in the categories Gr, Ab, and Top*. In Top an initial object is a and a final object is {a}. In Ri an initial object is Z, and the set with one point and the trivial ring structure, the so-called zero ring, is a final object. The mono- and epimorphism h : Z ---+ P, known from previous examples, is a right zero morphism but not a left zero morphism.

1.8 Diagrams In this section we want to make precise the notion of a diagram introduced in Section 1.2. Thus a diagram in a category %' will be a functor from a diagram scheme 9, that is, from a small category 9 (see Section 1.1)) into the category V. If the diagram scheme is finite, one says that the diagram is finite, and one illustrates the functor by its image. In this case we write down the objects in the image of the functor 9 and the morphisms as arrows between the objects. We omit the identities and often also morphisms which arise from other morphisms by composition. The commutativities which shall hold for all diagrams over the diagram scheme 9 are expressed by equality of morphisms in 9. Certainly, for certain diagrams additional parts may become commutative because of the particular properties of the objects and morphisms in the image of 9. Observe that the image of a functor, that is, the image of the map of objects and the maps of morphisms, does not form a category in general. In fact it is not necessary that all possible compositions of morphisms in the image are again in the image. For example, let 9 : 9 ---+ V be a functor with F A = F B for two different objects A, B E 9. Then two morphisms f : C --+ A and g : B + D cannot be composed in 9 but

The and thus 9 g F f is not necessarily contained in the image of 9. image of a functor 9, however, is a category if 9is an injective map on the class of objects. As in Section 2.1 we can form the category Funct(9, U). T h e objects of this category are diagrams. One also calls this category the diagram category. We observe that only the point of view differs from the one in Section 1.2. The category certainly is a functor category. I t is interesting to know how the morphisms between two diagrams can be illustrated. Let us clarify this with an example.

1.8

DIA.GRAMS

25

Let 9 be a category with three objects X , Y , 2 and six morphisms l , , I , , I , , x :X - t Y , y : Y 2, and x = yx :X + 2. Let 9and $9 be two diagrams and let cp : 9-+ $9 be a morphism of diagrams. --+

T h e n we can present all these data with the diagram

where all four quadrangles are cornmutative because y is a natural transformation. T h e category constructed here is also called the category of commutative triangles in V. T h e morphisms between diagrams are also families of morphisms, one for each pair of corresponding objects in two diagrams, such that these morphisms commute with the morphisms in the particular diagrams. Now let us take a fixed diagram in the sense of Section 1.2, which consists of a set of objects and morphisms, and let us ask the question whether this can be considered a diagram in the sense defined above. For that purpose, we form the subcategory 3i? of V with the same objects as given in the diagram and with all morphisms of V between them. 9’is a small category. Now let { L Z ~ ~ } ~be, , a family of small subcategories of 529, then d,, defined as the intersection of the corresponding sets of objects together with the intersection of the sets of morphisms, is a small subcategory of a. T h e composition is the one induced by 529. Let us choose for the 4.only those subcategories that contain all objects and morphisms of the given diagram. Then LZZi is the smallest subcategory of V which contains all objects and all morphisms of the diagram. Thereby the given diagram is completed by additional morphisms which occur as compositions of given morphisms or as identities. T h e small category we obtained in this way will be considered as the diagram scheme for our diagram. If the diagram scheme consists of two objects X and Y and of three morphisms 1, , 1 , and x : X + Y , then we call this category 2. T h e diagrams of Funct(2, V )are in one-one correspondence to the morphisms of V . Thus one calls Funct(2, V ) the morphism category of V. A mor-

niE,

n

1.

26

PRELIMINARY NOTIONS

phism in Funct(2, %) between two morphismsf : A is a commutative diagram

--f

B and g : C + D

A-C

fl

1.

B-D

1.9 Difference Kernels and Difference Cokernels As in Section 1.5, we want to generalize again a notion from S to arbitrary categories. For this purpose, let f : A --t B and g : A + B be two set maps in S.Then for f and g we can define a set C by C = {c I c E A

For an arbitrary object D Mor(D, C)

E

and f ( c )

S we consider

Mor(D,i)

= g(c)}

Mor(D,f)

Mor(D, A) A,Mor(D, B) Mor(D,g)

where i : C -+ A is the inclusion. Byfi

= gi,

we also have

Mor(D,f) Mor(D, i ) = Mor(D, g) Mor(D, i ) .

Conversely, if h E Mor(D, A) with Mor(D,f)(h) = Mor(D, g)(h),that is, = gh, thenf(h(d)) = g(h(d)) for all d E D. Thus all elements of the form h(d) are already in C, that is,

fh

h

=

(D-hLC A A )

or

h

=

Mor(D, i)(h‘)

Since i is injective and also Mor(D, i), we can use Mor(D, i) to identify Mor(D, C) with the set of morphisms in Mor(D, A) which are mapped onto the same morphism by Mor(D, f ) and Mor(D, g). We shall prove in a more general form that this property determines the set C and the injection i uniquely up to an isomorphism, as required for the generalization. We want to reformulate the conditions for the morphism sets. For each pair of morphisms (f,g) from A to B, we constructed a morphism i : C --+ A which satisfies the following condition: If D E S and h E Mor(D, A) and if fh = gh, then there is exactly one morphism h’ E Mor(D, C) such that h = ih’. Let %? be a category. Let f : A 4B and g : A + B be morphisms in V. A morphism i : C -+ A is called a dijference kernel of the pair (f,g)

1.9

DIFFERENCE KERNELS AND DIFFERENCE COKERNELS

27

if f i = gi and if to each object D E $'? and to each morphisms h : D + A with f h = gh, there is exactly one morphism h' : D ---f C with h = ih'. T h e morphisms considered form the following diagram:

D

LEMMA I . Each difference kernel is 61 monomorphism. Proof. Let i be a difference kernel o f ( f , g). Let h, k : D + C be given with ih = ik. Thenf(ih) = g(ih). Also by definition there is exactly one morphism h' : D -+ C with (ih) = ih'. But h as well as k comply with this condition. By uniqueness we get ,h = k.

LEMMA 2. If i : C + A and if : C' + A are dtference kernels of the pair (f,g), then there is a uniquely determined isomorphism k : C -+ C' such that i = i'k. Proof. Let us apply the fact that i is a difference kernel to the morphism i';then we obtain exactly one k' : C' C with if = ik'. Correspondingly, one obtains exactly one k : C + C' with i = i'k. Thus the uniqueness --+

of k is already proved. Furthermore, both assertions together imply

i = ik'k and if = i'kk'. Since i and i' are monomorphisms by Lemma 1, we get k'k

=

1, and kk'

=

1,.

.

I n the special case of S, this lemma proves also that if a morphism = gi' complies with the conditions on the diagram of the morphism sets, then i' can be composed with an isomorphism such that the composite is the morphism i. T h u s we get from the generalization of the notion given in the beginning only isomorphic sets with uniquely determined isomorphisms. Apart from that, the notion is preserved. Here we meet for the first time an example of the so-called universal problem. In the class of morphisms h with f h = gh the difference kernel i is universal in the sense that each h of this class may be factored through i : h = ih'. A category V is said to have difference kernels if there is a difference kernel to each pair of morphisms in 5f with common domain and range. We call %? a category with dzflerence kernels. Instead o f calling the mor-

if : C' + A with J;'

1.

28

PRELIMINARY NOTIONS

phism i a difference kernel, we often only call its domain C a difference kernel assuming that the corresponding morphism is known. We acted similarly in the case of subobjects. Since a difference kernel is a monomorphism, there is an equivalent monomorphism which is a subobject. This again is a difference kernel of the same pair of morphisms. Subobjects which are simultaneously a difference kernel of a pair of morphisms are called difference subobjects. Let V be a category with zero morphisms. Let f : A + B be a morphism in %'.A morphism g : C + A is called a kernel off if fg = O(C,B) and if to each morphism h : D -+ A with f h = O(D,B) there is exactly one morphism K : D + C with h = gh.

LEMMA 3. Let g be a kernel off. Theng is a dtzerence kernel of (f,O(A,B)). Proof.

fh

By the properties of the zero morphisms in V, we have that

= O(D,B)implies fh = f O ( D , A )and conversely. Thus the claim follows

directly from the definition. In particular, kernels are uniquely determined up to an isomorphism, and they form difference subobjects. Since the notions of a kernel and a difference kernel are different notions in general, the kernels which appear as subobjects get the name normal subobjects. Dually to the notions defined in this section we define dzyerence

cokernels, categories with dzference cokernels, difference quotient objects, cokernels, and normal quotient objects. For all theorems proved above, there are dual theorems. The difference kernel of a pair of morphisms (f, g) is denoted by Ker( f, g) and the difference cokernel by Cok( f, g). T h e kernel and cokernel of a morphism f will be denoted by Ker(f) and Cok(f) respectively. In all cases, we consider the given notations as objects in the given category and assume that the corresponding morphisms are known. Categories with difference kernels and difference cokernels are S, S*,Top, Top*,Gr, Ab, Ri, and RMod.We want to give the construction of a difference cokernel in S. Let two maps f, g : A --t B be given. Take the smallest equivalence relation on the set B under whichf(a) and g(a) are equivalent for all a E A. The equivalence classes of this equivalence relation form a set C, onto which B is mapped in the obvious way. This map is a difference cokernel of (f, g), as may easily be verified. Compare Problem 1.6 for the properties of Top. The properties of Top* arise analogously from the properties of S*.In Chapter 3 we shall deal with S*, Gr, Ab, Ri, and .Mod in more detail.

1.1 1

29

PRODUCTS AND COPRODUCTS

1.10 Sections and Retractions A morphism f : A -+ B in a category $7 is called a section if there is a morphism g in $7 such that gf = 1, . f is called a retraction if there is a morphism g in $7 such that f g = 1 B . I f f is a section with gf = 1, , then, by definition, g is a retraction and conversely. I n general each section determines several retractions, and conversely. T h e notions section and retraction are dual to each other.

LEMMA1. Each section is a diflerence kernel. Proof. Let f : A --t B be a section and g be a corresponding retraction. We show that f is a difference kernel of ( f g , I,). First, fgf = f = 1, f . Let h : C ---f B be given with fgh = I,h = h. Then by h = f ( g h ) the morphism h may be factored through f . If h = fh', thengh = gfh' = h', that is, the factorization is unique.

LEMMA 2. Let .F: V is a section in 9.

---f

9 be a functor and f be a section in V. Then 9 f

Proof. Let g be a retraction for f . 'Then gf

=

I,, so S g S f

=

lF,.

LEMMA 3. f : A + B is a section in the category V i j and only i j Mor,( f , C) : Mor,(B, C) Mor,(A, C) is surjective for all C E 2?. ---f

Proof. Let f be a section with a corresponding retraction g , and let h E Mor(A, C). Then h := h ( g f ) = (hg)f = Mor( f , C)(hg).Conversely, let Mor( f , C) be surjective for all C E V. For C = A, there is a g E Mor(B, A) with Mor( f , A ) ( g ) = I , , consequently f g = 1,. T h e assertion of this lemma is of special interest in view of the definition of a monomorphism or an epimorphism. When dualizing theorems on categories, be careful not to dualize also the notions used in S. I n S all injective maps are sections except the map 0 + A with A # a . All surjective set maps are r'etractions. I n Ab the map Z 3 n t+ 2n E Z is a kernel of the residue class homomorphism Z --t 2/22!; however, it is not a section. In fact, if g : Z + .Z were a corresponding retraction, then 2g(l) = 1 E h. But there is no such element g ( 1 ) in Z.

1.1 1 Products and Coproducts Another important notion in the category of sets is the notion of a product of two sets A and B. T h e product is the set of pairs A x B

= {(a,b)

I a CIA

and b E B }

30

1,

PRELIMINARY NOTIONS

Furthermore, there are maps p,:A x B 3 ( ( a , b ) - a ~ A

and

p,:A x B 3 ( a , b ) t + b ~ B

We want to investigate whether this notion can again be generalized in the desired way to morphism sets. First, one obtains for an arbitrary set C Mors(C, A x B ) g Mors(C, A ) x Mors(C, B) using the following applications. To h : C + A x B one assigns (p,h, p,h) E Mor(C, A ) x Mor(C, B), and to a pair ( f , g )E Mor(C, A ) x Mor(C, B ) one assigns the map C 3 c tt (f(c), g(c)) E A x B. Furthermore, there are maps Mor(C, A x B ) 3 h hp,h E Mor(C, A ) and Mor(C, A x B ) 3 h t+ p,h E Mor(C, B), which are transferred by the bijection given above into the maps ~ A) Mor(C, A ) x Mor(C, B ) 3 (f,g) w f Mor(C,

and Mor(C, A ) x Mor(C, B ) 3 (f,g) ++g E Mor(C, B ) I n this way the product and the corresponding maps p , and p , are transferred to the morphism sets up to isomorphisms. We shall prove in a more general context that this property characterizes products in S. T h e isomorphism of the morphism sets found above may be also expressed in the following way: T o each pair of maps f : C -+ A and g : C -+ B, there is exactly one map h : C + A x B such that f = p,h and g = g,h. Let V be a category, and let A , B E V be given. A triple ( A x B, p , ,p,) with A x B an object in V and p,:A x B+A

and

p,:A x B - t B

morphisms in V is said to be a product of A and B in V if to each object C E V and to each pair ( f,g) of morphisms with f : C -+ A andg : C +B, there is exactly one morphism h : C + A x B such that f = p,h and g = p,h. Then the morphisms form the following commutative diagram A

1.1 1 PRODUCTS

31

AND COPRODUCTS

T h e morphisms p , and p , are called projections. Often we write (f,g ) instead of h. If C = A x B, then ( p,, ,p,) = lAXB by the uniqueness of ( P A PB)' We generalize the notion of a prod.uct to an arbitrary family of objects in %? where I is a set. An ob.ject A, together with a family { p i : nit, A, + of morphisms. is called a product of the A, if to each object C E %? and to each family {f,: C -+ A,},,, of morphisms Ai such that fi = p,h for there is exactly one morphism h : C -+ all i E I . T h e morphisms pi are called projections again, and instead of h, we often write ( f,).As above we have ( p i ) = lnA,. If I is a finite set, A, and (fl ,...,fn) then we also write A, x -.. x A, instead of instead of ( fi).If I = 0 , then to eachL object C E V there must be exactly one morphism h from C into the empty product E. In this case, the conditions on the morphisms fiare empty. Thus this requirement says that E is a final object. Conversely each final object is also a product on an empty set of objects. 9

niG,

&,

niEI

LEMMA1. Let ( A , { p , } ) and (B, {qi}) be products of the family {A,},,, in V. Then there is a uniquely determined isomorphism k : A B such that pi = qik. --f

Proof. In the commutative diagram (for all i E I )

there is a unique k , because (B, {q,}) is a product, and a unique h, because ( A ,{ p i } ) is a product. hk as well as 1, make both left triangles commutative. ( A ,{ p , } ) being a product, this morphism must be unique; thus hk = 1, . Correspondingly, one has from both right triangles kh = 1,. This shows that the product in S is already uniquely determined up to an isomorphism by the condition on the morphism sets. Here we have another universal problem. For all families of morphisms into the particular factors with common domain, the product has the property that these families may be factored through the product with a uniquely determined morphism. Often we call product only the corresponding object of a product and assume that .the projections are known. If each [finite, nonempty] family of objects in %? has a product, then we call V a category, with Ifinite, nonempty] products. If ( A , { pil}) is a product of a

32

1.

PRELIMINARY NOTIONS

in V and if h : B + A is an isomorphism, then family of objects (B, { pih}) is another product for the A, .

LEMMA 2. Assume that in the category V there is a product for each pair of objects. Then V is a category with finite, nonempty products.

Proof. Let A, ,...,A, be a family of objects in V. We show that (-..(A, x A2) x x A, is a product of the A, . For an induction, it is sufficient to prove that (A, x .-.x A,-,) x A, is a product of the A,. Let p , : (A, x x A,-,) x A, -+ A, and q : (A, x

'**

x An-J x A,

+

A, x

'.'

x An-l

be the projections of the outer product and p i ( i = 1 , ..., n - 1 ) be the projections of the inner product. Let {f,}be a family of morphisms with common domain I3 and ranges A,. Then there is exactly one h : B -+A, x x A,-, through which the f i ( i = I, ..., n - 1) may be factored. For h and f , ,there is exactly one k: B + (A, x .-.x A,-,) x A, with qk = h andp,k =f,. Thenp,k =f , andpiqk =f,, i = l,..., n - 1 . The plq, ...,p,-,q, p , are the projections. k is uniquely determined by the given properties of the factorization. Similarly to the proof given above, one can also break up infinite products; specifically, one can split off a single factor by nAirA, xnAi iel

with

J u { j }= I and j $ J

i EJ

Thus, the product is independent of the order of the factors up to an isomorphism and is associative.

LEMMA3. Let be a family of objects in a category V, and let there be a product ( A , {pi}) for this family. pi is a retraction if and only if Mor,(A, , Ai) # ,a for all i E I and i # j .

Proof. Assume Mor,(A,, A,) # O . Then there is a family of morphisms f , : Aj -+ A, for all i E I with f j = l,, . The corresponding morphism f : A, -+ A has the property pi f = l A i . Conversely, let p , be a retraction with a section f : A, -+ A. Then p , f E Mor,(Aj , A,) for all i E I. The last lemma shows in particular that in a category with zero morphisms the projections of a product are always retractions. I n S the product of a nonempty set A with 0 is the empty set. Thus p , : 0 --t A cannot be a retraction. One easily shows that p , is not even an epimorphism.

1.12

INTERSECTIONS AND UNIONS

33

Let be a family of objects i.n a category V with A, = A for all i E I . Let B be the product of the A, with the projections pi . T h e identities 1, : A -+ A, induce a morphism d : A -+ B called the diagonal. A well-known example for this map is R 3 x ++(x, x) E R x R in S. T h e notions dual to the notions introduced up to now are coproduct with the corresponding injections, category with [finite, nonempty] coproducts, and codiugonal. T h e coproduct of a family {A,},,, will be denoted by A, , T h e product has lbeen defined in such a way that

for all B E V. Correspondingly, we have for coproducts

n

Morg(Ai , B ) ~6 Moryp(uA i, B )

for all B E V. In a more general context in Chapter 2, we shall study further properties of products and coproducts. T h e categories S, S*, Top, Top*, Gr, Ab, Ri, and .Mod are categories with products and coproducts. I n all these categories the products coincide with the set-theoretic products with the appropriate structure. T h e coproduct in S and Top is the disjoint union, in S* and Top* it is the union with identification of the distinguished points. I n Ab and in .Mod the finite coproducts coincide with the finite products. (Certainly this is only true for the corresponding objects. T h e injections are different from the projections, of course.) In Gr the coproducts are also called “free products.” T h e coproducts in Gr and Ri will be discussed in Chapter 3. We give another example from Chapter 3 without going into details about the definition. Let t7 be a commutative, associative, unitary ring. Let be the category of commutative, associative, unitary C algebras. I n .A1 the coproduct is the tensor product of algebras.

1.12 Intersectialns and Unions Let B be an object of a category %, and let f i : A, -+ B be a set of subobjects of B. A subobject f : A -+ B which is smaller than the subobjects Ai is called the intersection of the A, if for each C E V and each morphism g : C -+ B which may be factored through all A, (g = fihi) there is a morphism h : :C -+ A with g = f h . h is uniquely determined because f is a monomorphism. T h e intersection of the Ai will be denoted by A, . Letf’ : A’ -+B be a subobject which is larger than the subobjects A, . Let C E %, let g : B + C be a morphism in V, and let R : C‘ -+ C be a subobject such that g restricted to all the A, may be factored through k(gf, = Rh,). If these data always imply that the

34

1.

PRELIMINARY NOTIONS

morphism g restricted to A' may be factored through K (gf' = Kh), thenf' : A' -+B is called the union of the A, . Since k is a monomorphism, h is uniquely determined. T h e union of the A, will also be denoted by

u The A, intersection and the union of the A, are uniquely *

determined because the morphisms h in the definition of the intersection and the union are unique. This may be shown similarly to the proof of the uniqueness of the products u p to an isomorphism in Section 1.11, Lemma 1. One has to use two subobjects which fulfill the conditions given above, and one has to compare them by the unique factorizations. As subobjects they are not only isomorphic but equal. Since the subobjects form an ordered class, it is easy to show that the intersections as well as the unions are associative, if one observes that the intersection of a subfamily of subobjects is larger than the intersection of the whole family, and that the union of a subfamily is smaller than the union of the whole family. Observe that in the definition all objects of the category V are admitted as test objects, not only the subobjects of B. It may well be that B does not have sufficiently many subobjects to test whether another subobject is an intersection or union. If there is an intersection or a union for each [finite, nonempty] family of subobjects of each object, we call the category V a category with [$nit., nonempty] intersections or unions respectively. If V is a locally small category with finite intersections and unions, then the set of subobjects of each object in %' is a lattice. If there are arbitrary intersections and unions in V, then the subobjects of an object form a complete lattice, I n Chapter 2 we shall give more criteria for determining whether a category has intersections and unions; thus we do not give any examples here. Note that the notions intersection and union are not dual to each other. T h e corresponding dual notions are cointersection and counion. However, we shall not use these notions.

1.13 Images, Coimages, and Counterimages Let f : A -+ B be a morphism in a category V. T h e image off is the smallest subobject g : B' + B of B to which there exists a morphism h : A + B' with gh = f. Since g is a monomorphism, h is uniquely determined. If h is an epimorphism, then h is called the epimorphic image off. T h e image off is often denoted by Im( f ), where we assume that the morphism g is known and consider Im( f ) as an object. If there are [epimorphic] images for all morphisms in V, then we call V a category

1.13

IMAGES, COIMAGES, AND COUNTERIMAGES

35

with [epimorphic] images. Dually, wt: define [monomorphic] coimages and denote them by Coim( f ). If A’ is a subobject of A, then we denote the image of the morphism A’ -+ A -+ B byf(A’).

LEMMA1. If W is a locally small category with intersections, then V is a category with images. Proof. Form the intersection of all those subobjects of B through B may be factored. This intersection exists and is the which f : A smallest subobject with the property that f may be factored through it. -+

LEMMA 2. If W is a category with images and diference kernels then all images in W are epimorphic images. Im( f ) 2 B be a factorization off through its image, Proof. Let A and let k, k’ : Im( f ) -+ C be givein with kh = k’h. Then h may be factored as A -+Ker(k, k’) -+ Im( j).Since Ker(k, k’) -+ Im( f ) -+ B is a monomorphism and I m ( f ) is minimal Ker(k, k’) = Im( f ), thus k = k’ and h is an epimorphism. Let f : A -+ B be a morphism in V and g : B’ -+ B be a subobject of B. A subobject A’ -+ A of A is called a counterimage of B‘ under f if there is a morphismf‘ : A’ -+ B’ such that the diagram A’ f ’+ B‘ f

A-+B

is commutative and if for each commutative diagram B’

C -+

1

f

1.

A-tB

there is exactly one morphism h : C -+ A‘ such that the diagram C

36

1.

PRELIMINARY NOTIONS

is commutative. This condition asks for more than that A' be only the largest subobject of A which may be transferred by f into B'. But the condition implies this assertion. Thus the counterimage is also uniquely determined. For the counterimage of B' under f,we also writef-l(B'), neglecting the monomorphism f-'(B') --t A. Now we want to know which of the relations valid for the notionsf(A) and f - l ( A ) in S may be generalized. We collect the most important relations in the following theorem.

THEOREM. Let f : A --+ B and g : B + C be morphisms in V. Let A, C A, C A and B, C B, C B and C, C C be subobjects of A, B, and C, respectively. Then we have (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

(k)

f (A,) Cf (A,) if both sides are dejned. f-'(B,) Cf-l(B,) if both sides are defined. A, C f - ' f ( A l ) if the right side is defined. f f -l(B,) C B, if the left side is defined. f -I(B,) = h(B,) i f f is an isomorphism with the inverse morphism h. f-l(g-l(C,)) = (gf)-l(C,) if both sides are defined. g ( f (A,)) = (gf)(A,)i f both sides are defined, i f f (A,) a n d g ( f ( A 1 ) ) are epimorphic images, and if V is balanced. f ( A , ) = f f - ' f ( A l ) i f f - ' f ( A , ) is defined. f-l(B,) = f-Yf-'(Bl) if ff-'(B1) is defined. For each family of subobjects of A we have U f ( A i ) = f ( U A i ) if Ai is defined and %? is a category with images and coimages. For each family of subobjects {Bi}iezof B we have nf-'(Bi) = Bi) if the right side is defined.

u

f-l(n

Proof. The assertions (a)-(e) arise directly from the corresponding definitions. (f) We start with a commutative diagram

..

A

f

*B>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).

2.

80

ADJOINT FUNCTORS AND LIMITS

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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ADJOINT FUNCTORS AND LIMITS

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

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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.

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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

2.7

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

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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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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

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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

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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

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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.

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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.

<

4.6

Additive Functors

T h e facts that the morphism sets of an additive category V are additive groups and that the composition of morphisms is bilinear correspond to the condition for the functors 9 : %? -+ 9 between additive categories that for all A , B E %? the maps %(A, B ) : Homv(A, B )

--f

Hom2(S-A, .FB)

(1)

4.6

179

ADDITIVE FUNCTORS

are group homomorphisms. A functor 9 which satisfies condition (1) is called an additive functor. Of course, there are also other functors between additive categories which are not necessarily additive. Because of the bilinearity of the composition of morphisms in an additive category 9,the representable functor represented by any object A in $? is additive where we mean the functor Hom,(A, -) with values in Ab.

THEOREM 1. A functor 9 : % + 9 between additive categories is additive ;f and only ;f 9preserves finite direct sums with the corresponding injections and projections. Proof. If 9 is additive, then 9 preserves condition (2) of Section 4.2, Corollary 2 for direct sums. If 9 preserves finite direct sums with their f g ) = 9(f) 9 ( g ) . In fact, injections and projections then 9( let objects A , B, C, D in $? and morphisms f : A + B and g : B + D be given, then f @ g is uniquely determined by ( f @ g ) qA = qcf and ( f @ g ) qB = qDg. These conditions are preserved by 9. Furthermore, 9 preserves diagonals and codiagonals of finite direct sums. Hence by Section 4.1, Corollary 1 we have

+

F(f

+ g)

3(ffB g) g ( d A )

= 3.(vB)

+

=I v,B(g-f

@ Fg)

' F A

=

gf

+ Fg

For an abelian category we can also ask for the preservation of certain In the diagram exact sequences by the functor 9.

0

0

0

the sequence of the f i is exact if and only if the sequences 0 -+ B,-l -+ A i4Bi+ 0 are exact where Bi = Im(fi). Then this is equivalent to Bi-l = Ker(fi). If 9 preserves short exact sequences, then 9 also preserves arbitrary exact sequences. Thus we call 9 an exact functor if 9preserves short exact sequences. If 9preserves exact sequence of the form O-tA+B-tC

or

A+B-+C+O

180

4.

ABELIAN CATEGORIES

then S is called left or right exact respectively. If for each exact sequence 0 + A + B --+ C -+ 0 the sequence F A -+ .%B + S C is exact, then .% is called a half-exact functor. Be careful not to confuse the condition for a half-exact functor with the condition that for each exact sequence A + B + C , the sequence S A -+ S B + 9 C is also exact, since in this case .% is exact. A functor S is left or right exact if and only if .% preserves kernels or cokernels respectively. Obviously, each exact functor is left exact and right exact, and each left or right exact functor is half exact. Furthermore, a functor which is left and right exact is exact, as one can easily see by Section 4.3, Lemma 2.

PROPOSITION 1. A half-exact functor .% : V + 9 between abelian categories is additive. Proof. By Theorem 1 we only have to show that .% preserves direct sums of two objects with the corresponding injections and projections. If we characterize these by Sections 4.3, Lemma 3, then we obtain by the half exactness of S that .%piSqi = lFAi,and we obtain the exactness of the sequences

From the first condition, we may already conclude that the S p , are epimorphisms and the S q , are monomorphisms. Thus the sequences

are exact by Section 4.3, Lemma 2. This proves the proposition by Section 4.3, Lemma 3. An example of a left-exact functor from an abelian category V into the category Ab is again the functor Hom,(A, -) : V + Ab represented by an object A E W. I n fact, if 0 -+ B +!- C 2 D is exact and h : A + B is a morphism with Hom,(A,f)(h) = 0, then f h = 0. Since f is a monomorphism, we have h = 0; hence Hom,(A,f) is a monomorphism. If h’ : A + C is a morphism with Home(A,g)(h’) = gh’ = 0, then there exists a morphism h : A + B with f h = h’ because f : B -+ C is

4.7

181

GROTHENDIECK CATEGORIES

the kernel of g . Consequently, Hom,(A,f)(h) = h'. Together with Hom,(A, g ) Hom,(A, f ) = 0, this implies the exactness of the sequence Ffomy(A.f)

0 + Homu(A, B ) -A HomV(A, C )

HOrny(A.o)

Homv(A, D)

4.7 Grothendieck Categories Let ti be a small category. Then the functors from & into the abelian category 5' together with the natural transformations form a category Funct(8, V).

PROPOSITION I . Funct(8, V) is an abelian category. Proof. By Section 2.7, Theorem 1, Funct(8, V ) is finitely complete and cocomplete. Furthermore, the functor 0 : B'+ V with O(E) = 0 for all E E 8 is a zero object for Funct(6, V). As in Section 4.1, we can define a morphism 6 from the coproducts into the products. Then for the functors FiE I:unct(8,%') the morphism 6,(E) : Si(E) -+ Ti(,?) coincides with a,(,) , that is, 6 is formed argumentwise. Hence by Section 1.5, 6 is an isomorphism in Funct(8, V). Correspondingly, d and C have to be formed argumentwise. Furthermore, the morphisms s ~ , ( are ~ ) natural transformations satisfying condition (4) for additive categories. T h e natural transformation h of Section 4.2 from the coimage into the image of a morphism in Funct(8, V ) is also formed argumentwise. Thus h is always an isomorphism and Funct(&,V) is an abelian category.

n

Since by Section 2.7, Corollary 2 the colimits commute with coproducts and cokernels, we obtain the following corollary. : Funct(8, %') -+ V is right exact COROLLARY 1. The functor lim -+ exists.

if

it

In the following let 5' be an abelian cocomplete category, Furthermore, we require that a certain condition holds in V which holds in all module categories. For each subobject 13 C A and each chain of subobjects {Ai}of A,

(uA') n B u ( A n~B ) =

(1)

holds. This condition is called the Grothendieck condition. Observe that

4. ABELIAN

182

CATEGORIES

Equation (1) does not hold for arbitrary sets {Ai}of subobjects of A in module categories. An abelian, cocomplete, locally small category with the Grothendieck condition will be called a Grothendieck category. In the following we shall need condition (1) not only for chains of subobjects of A , but also for directed families of subobjects. Here we mean by a directed family of subobjects of A a functor 9from a directed small category 8 into the category V such that F ( E ) is a subobject of A for all E E 8 and such that for all E -+ E‘ in € the morphisms S ( E )-+ S ( E ’ ) together with the monomorphisms into A form a commutative diagram F(E) ’9 ( E ‘ )

A This means that there is a natural transformation p : F --t X, from the functor 9 into the constant functor X, : 8 -+V such that p ( E ) : 9 ( E )-+ X A ( E )is a monomorphism for all E E 8.

LEMMA 1. Let 9 be an ordered set in which for each subset {vi} there exists a supremum vi . Let be a subset of 23 which is closed with respect to forming suprema in 9 of chains in m. Let 0 # 9‘ C 9. If then UaEB,v $m, then there are already Jinitely many vl ,..., v, E %’ with v1 v * . - v v, $ m.

u

Proof. Let p(%’)be the power set of ‘23’. Each subset of 2l may be well-ordered in different ways (independent of the given order in 9). Let Q(9’)be the set of all well-orderings of all subsets of %’. Thus each element of Q(9’) has an ordinal number. Let a’(%’) be the subset of those elements of Q(9’)for whose corresponding set P E p(%’)we have U r E Pv $2B. By hypothesis, a’(%’)is not empty; thus there exists a Q E a’(%’) with smallest ordinal number y. Let P E Q(B’) be the corresponding subset of % and assume that the elements of P have as subscripts ordinal numbers smaller than y in the order of the given well-ordering. Then for all p < y we get u , < o Urn Em. Hence, u a < v u a < o 0, v, because is closed with respect to suprema of chains. T h e set of the v, is, in fact, a chain. Hence, y cannot be a limit. If y is infinite, then there is a bijection between the ordinal numbers smaller than y and the ordinal numbers smaller than y - 1. This bijection maps y - I to 0 and n to n 1. This reordering does not change the value of (J1,EP v. This is a contradiction to the minimality of y. Consequently, y is finite.

+ urn,,

9

+

LEMMA 2. Let V be a Grothendieck category. Let B C A be a subobject of A E V and let {Ai}be a directed family of subobjects of A.

4.7 Then

183

GROTHENDIECK CATEGORIES

(u

A,) n B

=

u

(A*n B )

(u

Pro,/. Since A , n B C Ai)n B , we get in general U (Ai n B ) C A,) n R. Let C = (A, n B). T h e set of the subobjects of A forms an ordered set with suprema. We define a subset of the subobjects of A by D E!~X if and only if D n B C C. By the Grothendieck condition, is closed with respect to suprema of chains. Assume that (U Ai)n B $ C. Then, by Lemma 1, there exist A , ,..., A, with ( A , u -..u A,) n B $ C . Since the {Ai}form a directed family of subobjects, there exists an A, with A j C A, for j = 1,..., n. Hence, A, n B $ C. Obviously, this is a contradiction. Consequently, (U A,) n B = (Ain B). After having extended the Grothendieck condition to directed families of subobjects we now want to discuss the importance for direct limits. For this purpose, let V be a Grothendieck category, 8 be a small directed category, and 9 : 8 -+ %? be a functor. We denote the objects in € by i, j , k , ..., and set F(i)=: Fi . For i < j we denote the morphism from Fi into F j induced by 9by fii : Fi-+Fi . T h e injection will be denoted by qi : Fi lim 9. ---t

u

(u

u

-+

LEMMA3 . Let V be a Grothendieck category and € be a directed small category. Let 9E Funct(8, V). Then we have Ker(q, : Fi

---f

u

lirn 9) = Ker(fi, : F, +F j ) + i<j

Proo/. We denote Ker(q,) by K i and I<er(fii) by Kii commutativity of

Fi

. Because

of the

*Fj

lirn 5 4

uiGj

<

for all i j we get Kii L K i , hence Kii C Ki . By Section 2.6, Proposition 2 lim 9 is the cokernel of g : UiGiFii + 9 Fi where Fti = F, for all i and where g is defined componentwise for the Fij by g i j = qi -- qjfij . Let Aij = Im(gii). Then the sequence

<

O+

u

A,, +

UF,

--t

I4 imP-+O

is exact. Let E be a finite subset of the set of pairs (i,j ) with i < j and i, j E 8 and let A,

=

u

(n.,)eE

A,,

184

4.

ABELIAN CATEGORIES

The set of finite sets E defined in this way together with the relation of containment forms again a directed small category 8'. We get U i G i Atj = U E E d A, * T h e diagram

is commutative with exact lower row. Since (Kk -+ there exists exactly one morphism Kk-+ Aij with

u

then there exists a unique morphism

(uAij)

A Fk-+

Fi 1Lm F)= 0, --f

Kkwith

This implies that ((J Aii) nFk = Kk as subobjects of Fi. For E E 8' let 1 2 j for all j with (i, j ) E E and 1 2 k. Furthermore, 1, and hi, = 0 define morphisms hi, : Fi-+F, by hi,= filfor i otherwise. This defines a morphism h : JJF i+F,. T h e diagram

<

4.7

185

GROTHENDIECK CATEGORIES

is commutative and we get ( A , -+ Fi --t F,) = 0 because (Fii + IJ Fi -+ F , ) = 0 for j 1 by definition of h. Hence, (A, n Fk -+ F,) = 0. Since K k l = Ker(Fk -+ F,) and A, n Fk is a subobject of Fk , we have A, nFk C Kkl . This proves

<

&-=(UA,,)nF,.=

(AEnFk)CUKkl k,(l

where we used Lemma 2.

THEOREM 1 . Let V bu an abelian, cocomplete, locally small category. The following assertions are equivalent:

( I ) Direct limits in V are exact. (2) For each directed family (Ai),E,of subobjects of A E V , the morphism

lim(Ai) -+ A is a monomorphism. -+ ( 3 ) 9 is a Grothendieck category. Proof. (1) 3 ( 2 ) : We consider I as a directed small category and form the functors 9 : I + V with 9 ( i )= Ai and 3 : I --+V with 3(i)= A for all i. T h e morphisms o f I are mapped into the monomorphisms of the Ai and A into A respectively. Since lim 3 = A , (1) implies that 9-+ A is a monomorphism. lim -+ (2) 3 (3): Let {A,} be a chain of subobjects of A and B C A. By the second isomorphism theorem, we get a commutative diagram with exact rows and columns 0

0

0

1

1

1

O-AinB-

-+

1 O-A,/A,n

1 1 0

B-+

B

- - +

B/Ai n B -0

1

1

1 1

I

0

0

A , v B / B -0-0

for all i E I. Morphisms A i-+ Ai induce morphisms between the corresponding 3 x 3 diagrams such that all occuring squares are commutative.

4.

186

ABELIAN CATEGORIES

If we apply the functor I& to this chain of 3 x 3 diagrams, then we get that lLm(A, n B ) + A , B -+ A , lim A, -+ A, and I$(A, U B ) -+ A are monomorphisms by (2). Since A,-+ A may be factored through I& A, and A, -+ 1Lm A, is an epimorphism we may identify 1Lm A, with A, as subobjects of A. Since ILm is right exact and preserves

u

isomorphisms, the diagram 0

0

0

I 1 1 1 -u - 1 I 1 -1

0

0

0

lim(B/A, n B ) +

0

u ( A ,u B )

lim(A, u B/A,) -+

0

lim(A, u BIB)

0

0

+B

u(A,nB)

Ai

lim(Ai/Ai n B ) +

+

1

1

0

0

0

is commutative with exact rows and columns. Now let morphisms D + B and D -+ A i with

u

be given.Then (D-+ l$(B/A, n B ) ) = 0 and ( D -+ li$(Ai/A, n B ) ) = 0. Hence, there exist f : D -+ ( A in B ) with

u

( D + B ) = (D f U ( A i n B ) + B ) and g : D + (J(A, n B )

u

u

uu

(D A,) = (D5 (Ain B ) -+ A,). However, since (D1, ( A n~ B ) + ( A , u B ) ) = (D5 ( A , n B ) -+ (A,u B)) with

u

-+

u

u

u

u

and since ( A , n B ) -+ ( A , u B ) is a monomorphism, we getf = g. Consequently, ( A , n B ) is the fiber product of A, and B over (J ( A iu B), hence (A,nB) = A,) n B. (3) => (1): By Corollary 1 and Section 4.3, Lemma 2 it is sufficient to show that for a directed category Q and a natural monomorphism

u

u

(u

u

4.7

GROTHENDIECK CATEGORIES

187

p : 9+ 29 also lim p : 1 59-+ 1Lm $9 is a monomorphism. Since

Funct(&, V) is abelian and kernels are formed argumentwise in Funct(8, V), the morphisms p ( i ) : 9 ( i )-+ "(i) are monomorphisms for all i E 8. We denote 9 ( i ) = Fi and g(i) = G, . Let K , = Ker(Fi .+ lim a),L, -= Ker(G, -+ l$ $9), A, = Im(F, -+ 1% F), --* B, = Im(Gi -+ 1 529), K = Ker(lLm p), and C be the fiber product of A, n K with F, over A , , Then we get a commutative diagram

Ki

1

0

----f

-

Li

1

1 1 1 1 - 19 1 9

K

lim

--f

+

lim +

u

where the last row is exact. Because of A, = 1 59and the Grothendieck condition, we get K = Ai) n K = ( A , n K ) , for the A , form a directed family of subobjects of 1@ F. If all A in K = 0, then also K = 0 and l i z p is a monomorphism, which we had to show. Since Fi -+ A, is an epimorphism, C -+ A, n K is an epimorphism by Section 4.3, Lemma 4. It is sufficient to show that this epimorphism is a zero morphism. We have (C -+ lim 29) = 0. Since C -+ Gi is a monomorphism, we get C C L , as subobject of G,. By Lemma 3 Li = UiGjLij where Lij = Ker(Gi -+ Gj), hence C = C n Li= Lij) n C = ( L i j n C). T h e diagram

(u

(u

u

u

I

1 1 Fi

-+

is commutative. Since (Lij n C - t Gj)

=

Gi 0 and Fi -+Gj is a mono-

188

4.

ABELIAN CATEGORIES

morphism, we get (Lii n C +Fj) = 0, hence Lij n C C K i j . Consequently, C = (J (Lij n C ) C Kij = Ki. Then the preceeding diagram implies (C +F, -+ A,) = 0. Since A, n K + Ai is a monomorphism, we get (C -+ A, n K ) = 0.

u

' be a Grothendieck category; then the morphism COROLLARY 2. Let % 6 : A, + A, (in Section 4.1) is a monomorphism.

n

Proof. Let We define A ,

=

be a set of objects in V. Let E be a finite subset of I. Since

&, A,.

we get that A, is a subobject of A, . T h e set of these subobjects forms A, = A i . Hence, a directed family of subobjects and we have A, = 1 9A,. On the other hand, the A , are also subobjects of A,. Thus, 1% A, + A, is a monomorphism.

u

n

n

As in the preceeding cases, we used here also a method which is typical for proofs in Grothendieck categories. We replace an infinite arbitrary union A, = A, by a union of a directed family of subobjects which all are finite unions. T h e corresponding conclusion in module categories, where the unions are sums (not necessarily direct sums) and where one can compute with elements, is that for

u

there is a finite index set il

,..., in with x E Ail

+ ... + Ain

,

COROLLARY 3. Let monomorphisms pi ; A, -+ B, be given and assume that A,, A, and M B, , Biexist in the Grothendieck category V. Then the morphism p : M A, -+ M B, induced by the p, is a monomorphism.

n

n

Proof. There is a commutative diagram

as in Section 4.1. By Section 2.6, Corollary 5 , products preserve mono-

4.7

189

GROTHENDIECK CATEGORIES

n

n Bi

morphisms. Since Ai Ai -+ Biis also a monomorphism. Ai -+ --f

is

a

monomorphism,

COROLLARY 4. Let 59 be a Grothendieck category. Let {Ai} be a directed A be a morphism in V. Then family of subobjects of A and f : B

Proof.

Let Im( f ) = A'. T h e commutative diagram f -'(Ai)+Ai n A'

1 B-

-

A,

1

+A'-A

and Section 2.6, Lemma 3 imply that the left square is a fiber product. Ai n A' is an epimorBy Section 4.3, Lemma 4(c) we get thatf-'(Ai) phism. Furthermore,f-'(Ai) contains the kernel K off. With the 3 x 3 lemma we get a commutative diagram with exact rows and columns --f

0

0

1 1

1

0

1 40-0 1

O - K - h K

1

1

0

- -Ai n A'

A'

A'/Ai n A'

1

1

1

0

0

0

-

0

In particular, the morphism B/f -'(Ai) -+ A'/Ai n A' induced by the epimorphism B -+ A' is an isomorphism. Hence, we also have B/f -I( Ai) A'/(U Ai) n A'. Since direct limits are exact, we get by the application of a direct limit to the above diagram again a corresponding commutative diagram with exact rows and columns. This implies an isomorphism B/Uf-'(Ai) g A ' / u (Ai n A'). By the Grothendieck condition, we have A'/U (Ain A') = A'/(U A$) n A'. Consequently, B/Uf -'(Ai)= B/f-'( U Ai) and Uf-'(Ai) = f - l ( U Ai).

u

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4.8 The Krull-Remak-Schmidt-Azumaya Theorem In Section 4.5 we investigated the uniqueness of sufficiently fine chains of subobjects of an object. Each decomposition of an object into a coproduct induces chains of subobjects in different ways. These, however, are not fine enough in the general case to allow the application of the Jordan-Holder theorem, even if we make the decomposition as fine as possible. Thus we shall use different methods to investigate the uniqueness of sufficiently fine decompositions into coproducts. Here we shall also admit infinite decompositions. In this section let V be a Grothendieck category. An object A E V is indecomposable if A # 0 and if, for each decomposition of A into a direct sum A = A , 0A , , either A = A, or A = A,. If A is not indecomposable, then A is said to be decomposable. An element r of a unitary associative ring R is called a nonunit, if Rr # R and rR # R. This is equivalent to saying that there is neither an x E R with xr = 1 nor a y E R with ry = 1. R # 0 is called a local ring if each sum of two nonunits in R is again a nonunit. An element r E R is called idempotent if r2 = r .

LEMMA 1. Let r be an idempotent in a local ring R. Then either r = 1.

=

0 or

r

Proof. We have (1

+

- r)2 = 1 - Y . Since 1 = (1 - r ) r is not a nonunit, either r or 1 - r is not a nonunit. If r is not a nonunit, then xr = 1 , hence r = xr2 = xr = 1. Symmetrically YX = 1 implies r = 1. If x(l - r ) = 1 and (1 - r)x = 1, then 1 - r = 1, hence r = 0.

An element r

E

R is called a unit if there is an x E R with XY

= YX =

1.

LEMMA 2. Let R be a local ring. Then the nonunits form an ideal N.All elements in R which are not in N are units.

Proof. Let r be a nonunit and x E R. We have to show that xr is a nonunit. By definition, there cannot exist a y E R with yxr = 1. However, if xry = 1, then (yxr)(yxr) = y(xry) xr = yxr. Sinceyxr is an idempotent, we haveyxr = 1. In fact ifyxr = 0, then 1 = (xry)(xry) = xr(yxr)y = 0, hence R = 0, a case which we want to exclude. Consequently, N is closed with respect to addition and multiplication with ring elements from both sides. If r E R and r N , then there is an x E R with xr = 1 or YX = 1. Assume xr = 1 . As above, we get (rx), = rx, hence YX = 1. r is a unit.

4.8

THE KRULL-REMAK-SCHMIDT-AZUMAYA

THEOREM

191

LEMMA 3. Let A E %?uith local endomorphism ring. Then A is indecomposable. If A is indecomposable and of finite length, then the endomorphism ring of A is local. Proof. Let A = B 0C and f = ( A -+ B - t A ) , where p : A -+ B is the projection and q B -t A the injection with respect to the decomposition into the direct sum. Then f = qpqp = f = 0 or f = 1. Hence, either B = 0 or B = A. Since the endomorphism ring of A is not the zero ring, we get A = 0, hence A is indecomposable.

Let A be indecomposable and of finite length. Let f : A -+ A be given. Then Ker( f ) C Ker( f 2, C is an ascending chain of subobjects of A. T h e commutative diagram 0

0

- -- - -- Ker(f2)

A

P"

Im(jz)

0

Ker(f)

A

P*

Im(f)

0

A with exact rows implies that Im( f ) 2 Im(f2) 2 for q*q"p" is the unique factorization off through Im( f 2). Both chains become stable after n steps, that is, we get Ker(f") = Ker( f n+r) and Im( f ") = Im(f?l+r) for all r E N, because A is of finite length. Let g = f". Let qp = g be the factorization of g through Im(g) and q'p' = g2 be the factorization of g2 through Im(g2). Since Im(g) = Im(g2), we get q = q'. We get a commutative diagram . * a ,

In fact, we havegpp

=

ppg. Since

(Ker(g)

--f

A

A

--f

Im(g2)) = 0

there exists a uniqueg' withg'p = pg = p'. qg'p = gqp implies qg' = gq, since p is an epimorphism. T h e fact that p' is an epimorphism implies that also g' is an epimorphism. Since Im(g) and Im(g2) have equal length, g' is an isomorphism with the inverse morphism h.

4.

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ABELIAN CATEGORIES

Then

Hence, qhp is an idempotent with image Im(g). By Section 4.3, Lemma 3, we get A = Im(g) @ Ker(qhp). Since A is indecomposable, either Im(g) = A or Im(g) = 0. In the first case, g and also f are isomorphisms because A is of finite length. I n the second case f = 0. Consequently, in the endomorphism ring Homw(A,A ) each element f which is not a unit, that is, which is not an isomorphism, is nilpotent, that is, there is an n E N with f = 0. Let f andf be nonunits. We assume that f f is not a nonunit. Then there exists an x E Hom,(A, A)with xf xf = 1. Since xf and xf are not units, they are nilpotent. Let i be minimal with (xf)i = 0 and let j be minimal with (xf ')j (xf)i-l = 0. i and j are necessarily different from zero. Then

+

+

(xj')j-l(xf)i-l = (xf')j-'(xf' =

(xf')j(xf)i-'

+ xf)(xf)"-l + (xf')j-'(.)Ef)i = 0

contradicting the minimality of i and j . Hence f Hom,(A, A) is a local ring.

+f

is a nonunit and

If, in the following, we talk about coproducts of subobjects of an object in V, then the monomorphisms which belong to the subobjects are assumed to be the injections of the coproduct. Nonequivalent subobjects, even if they are isomorphic as objects, will be denoted differently. T h e projections into the direct summands, however, may change without us changing the notation for the object which could be considered as a quotient object with respect to the projection.

LEMMA 4. Let A = A, and let the endomorphism rings Hom,(A, , A,) be local. Let f and g be endomorphisms of A with f g = 1, . Let E = {i, ,..., in}be ajinite subset of I . Then there exist subobjects B, ,..., B, of A and isomorphisms hj : A,, + B j for j = 1, ..., n such that for each j the diagram

+

A-A

h

4.8

THEOREM

THE KRULL-REMAK-SCHMIDT-AZUMAYA

193

is commutative for h = for for h = g . Furthermore, there exists a decomposition A = Bl @ @ B , @ Ai i$E

Proof. T h e injections and projections of the Ai will be denoted by qi and pi respectively. We have pifqi pigqi = l A i for i E I . Since Homv(Ai, Ai) is a local ring, one of the two summands, e.g., pifqi , is an automorphism with the inverse morphism ai : Ai -+ A i . Let i = i, E E. We factor fqil : Ail -+ A through B, := Im(fqil) as fqil = ql‘h, with q,‘ : B, -+ A and h, : Ail -+ B, . Sincepi,ql’h, = pilfqil is a monomorphism, h, is the isomorphism we were looking for. Furthermore, (ql’hlail = ql’hlailPil and Im(ql’hlailPil) = B1,

+

Ai

Ker(q,’k,ai,pil) = K e r ( p i l ) = i#il

By Section 4.3, Lemma 3 we have A = B, @ Hi+,, A,. Starting with this coproduct we now may replace AiP by B, . Then after n steps the lemma is proved.

THEOREM (Krull-Remak-Schmidt-Azumaya). category and A E 59.Let A

Ai

=

Let %? be a Grothendieck

with local rings Homy(A,,

Ai)

if1

and A

=

11 B j

wilh indecomposable B,

3t- J

be given. Then there exists a bijection y : I have Ai B,ci,.

--f

J such that for all i E I

we

Proof. T h e injections and projections of the Ai will be denoted by qi and p i respectively, those of the Bi by qj’ and pi’ respectively. First, we show that to each B j there exists an isomorphic Ai and that this isomorphism is induced by qj’pj’ : A -+ A. Then also the endomorphism rings of the Bj are local, the Ai are indecomposable, and the hypotheses of the theorem are symmetric. Let f = qj‘pjf and f ’ = 1 - f. Then f + f ’ = 1 , f 2 = j , and f f 2 = f‘. Furthermore, I m ( f ) = Ker(f’) = B j by Section 4.3, Lemma 3. Let E C I be a finite subset and A, = OirE Ai . T h e A, form a directed family of subobjects of A. There exists an E with since A, n Bi + 0, Bj = A,) n B j = (A, n B j ) . Let

(u

u

194

4.

ABELIAN CATEGORIES

E = {i, ,..., in}. By Lemma 4, there exist c k c A and isomorphisms hk : Aik-+c k , k = l,,.., n, which are induced by f or by f We assume that all h, are induced byf’. Let C, = c k . Then we get a commuI.

tative diagram A,nBi-A,gCE

1

1 1 f‘

9j’ Bi-A-+A

Since f ’qi’ = 0 and A, n Bi + A, C, + A is a monomorphism we get a contradiction to A, n Bj # 0. Hence there exists at least one io E E such that the square

A

f

-+

A

is commutative. By definition off, we get C,, C Bj . But since c k o is a direct summand of A , c k o is also a direct summand of Bj . Bi is indecomposable, hence Ck0= Bi . fqi, = qj)j‘qio implies hko = p j ’ q i o . Now let Aio B i o . We have to compare the number 1 a I of the A, isomorphic to Aio with the number I /I I of the Bj isomorphic to Bio . By symmetry, it is sufficient to show that I a I 2 I /I 1. First, assume that 1 0 1 I is finite. By the preceeding construction there exists corresponding to j , E P an i, ~a such that f, = qj1p;, induces an isomorphism Ai 1 -N Bjl . Furthermore,

Now, if we compare the second direct sum with A the same procedure, then we get after n steps

=

uic,A, and apply

Hcre we have to observe again that the injections of the Ai and Bi remain unchanged but that the projections change. T h e fact that the injections of the Ai remain unchanged also guarantees that the il ,...,in are all pairwise distinct because for a decomposition into a direct sum, no two injections can be equal. i, ,..., in E a implies now I a I 2 I /I I.

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INJECTIVE A N D PROJECTIVE OBJECTS A N D HULLS

195

Now let I a 1 be infinite. Let E C J be a finite subset, let j E , ! Iand j 4 E. Further, assume that A, B j by the isomorphism induced byf = qj'pj'. Then the diagram Ai n BE---tAi

Bj

1 1

3.

f

B,-AdA

is commutative where BE = Q j c E Bj . We get A, n BE = 0 because (BE+ A + A ) = 0 and A, n BE+ A, g Bj + A is a monomorphism. On the other hand ~i

=

(2

BE)

n Ai

=

u

ESJ

(

niB E )

~

+0

Hence there exists a finite subset E C J with Ai n BE # 0. Each j E J which induces an isomorphism Ai B, by qilpi' for the above determined i must lie in this E . Hence there are only finitely many such j . We call this number E(i). T o each j E /3 we may construct such an i. Hence,

u

E(i) = B

i€a

This proves I a

4.9

1 3 1 fl I. Injective and Projective Objects and Hulls

Let %? be an abelian category. An object P E %? is called projective if the functor Hom,(P, -) is exact. Dually, an object Q E $? is called injectiue if the functor Horn,(-, Q) is exact. Since the functor Hom,(A, -) is left exact for each A E 59,P is projective if and only if Hom,(P, -) preserves epimorphisms, that is, if for each exact sequence A 4 B + 0 and for each morphism f : P + B there is a morphism g : P-+ A such that the diagram P

J If is commutative. Dually, Q is injective if and only if for each exact

4. ABELIAN

196

CATEGORIES

sequence 0 -+ A -+ B and each morphism f : A -+ Q there is a morphism g : B -+ Q such that the diagram 0 --+

A

-+

B

Q is commutative. Since in a module category all epimorphisms are surjective morphisms, the projective modules in a module category coincide with the relatively projective modules introduced in Section 3.4 with respect to the forgetful functor into the category of sets.

LEMMA I . Let Pi E % and P if all Pi are projective.

=

JJPi begiven. P is projective i f and only

Proof. Let A-+ B -+ 0 be exact. Let morphisms f i : Pi --+ B and f :P - + B with qif = fibe given. We use the diagram

A-B-0

If P is projective, then there exists P ---f A with ( P B ) = ( P -+ A B). Hence, for each i, we obtain (Pi -+ P --t A -+ B ) = f i . Sincef is uniquely determined by the f i , all Pi are projective. Let the Pi be projective, then there exist Pi-+ A, making the above -+

-+

diagram commutative. In a unique way, these determine a morphism P -+ A with ( P -+ A -+ B ) = f . Hence, P is projective.

LEMMA 2. P is projective ;f and only i f each epimorphism A

-+

P is

a retraction.

Proof. Let P be projective and A morphism g in

s/ A-P-0

is a section for A

-+

P.

-+

f p

P be an epimorphism. Then the

4.9 Let A product

+

INJECTIVE AND PROJECTIVE OBJECTS AND HULLS

197

B -+ 0 be exact and P + B be given. We form the fiber c-P

A-B

Since A + B is an epimorphism, C + P is an epimorphism, hence a retraction with section P -+ C. Consequently, ( P - t B)

=

(P+ C + A + B)

A monomorphisni A + B is called an essential extension of A if each morphisni B - t C, for which A + B - t C is a monomorphism, is a monomorphism itself. A subobject A of B is called large if for each nonzero subobject c' of B also A n C is nonzero. LEMMA3. A subobject of B. Proof.

+B

and only ij A is a large

is an essential extension

-

-

We use the commutative diagram with exact rows 0 --+

A nC

1

A

1

+AIA

nC

1

0

0-C-B-D-0

where, as in the proof of the second isomorphism theorem, the vertical morphisms are monomorphisms. If A -+ B is an essential extension, and C 0, then B + D is not a monomorphism, nor is A -+ D a monomorphism. Hence, A n C = Ker(A -+ D) # 0 (see Section 4.4( 1)). If A is large in B, and B -+ D' is not a monomorphism, and D is the image of B -+ D',then C # 0, hence also A n C = I<er(A --t D) # 0.

+

COROLLARY 1. (a) A n essential extension of an essential extension is essential. (b) Let A + B be a monomorphism in a Grothendieck category and {Ci} be a chain of subobjects of B, all containing A. If all Ci are essential extensionf of A, then also (J Ciis an essential extension of A. Proof. (a) If A C B is large, and B C C is large, and 0 # D C C, then A n D = A n ( B n D) # O . (b) Let 0 j - D _C (J C i . Then D = Ci)n D = (J (Ci n D). For some i, we have Ci n D # 0. Hence, A n D = A n Cin D # 0.

(u

4.

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ABELIAN CATEGORIES

Since in a Grothendieck category 1% Ci = (J Cifor a chain {CJ, and 5Ciare monosince by Section 4.7, Lemma 3 , the morphisms Ci -+ 1 morphisms, in the preceding corollary the assumption that B exists is superfluous because B can always be replaced by lim Ci . A monomorphism A -+Q with an injective ozject Q is called an injective extension of A . An injective, essential extension is called an injective hull (injective envelope). An essential extension A -+ B will be called maximal if for each essential extension A -+ C, which may be factored through B (A-,C) = ( A - + B - + C ) the morphism B .+ C is an isomorphism. An essential extension A -+ B is callcd a largest essential extension if A -+ B may be factored through each essential extension A -+ C ( A - + B )= ( A + C + B )

An injective extension A -+ B is called minimal if for each factorization A -+ C -+ B of A -+ B with an injective object C and a monomorphism C -+ B the morphism C -+ B is an isomorphism. An injective extension A + B is called a smallest injective extension if for each injective extension A -+ C, there exists a monomorphism B --t C with ( A 3 C) = ( A -+ B C). -+

PROPOSITION 1 . Let A E % ' ? and assume that A has an injective hull. The following are equivalent for a monomorphism A -+ B : (1) A (2) A (3) A

B B -+ B (4) A + B ( 5 ) A -+ B Proof.

.+

-+

is an injective hull of A . is a maximal essential extension of A . is a largest essential extension of A . is a minimal injective extension of A . is a smallest injective extension of A .

We shall use the diagrams A-B

f

f

A-+B

C

C (1)

(2)

( 1 ) o (2): Let f be an injective hull andg an essential extension in ( 1 ) . Since f is essential and g is a monomorphism, h is a monomorphism.

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INJECTIVE A N D PROJECTIVE OBJECTS A N D HULLS

199

Since B is injective, h is a section: C = B @ D . Since g is essential, D = 0, hence h is an isomorphism. Conversely, let f be a maximal essential extension and g be an injective hull of A in (1). h exists because C is injective, and is a monomorphism because g is a monomorphism and f is essential. Since f is maximal, h is an isomorphism and B is injective. (1) o (3): If in (2), f is an injective hull and g an essential extension, then there exists h because B is injective. Conversely, let f be a largest essential extension of A and g be an injective hull of A in (2), then h is a monomorphism because g is essential and f is a monomorphism. C being injective implies that h is a section, hence an isomorphism. (1) ci (4): If in (2), f is an injective hull, g an injective extension, and h a monomorphism, then h is a section, hence an isomorphism. Conversely, if in (2), f is a minimal injective extension and g an injective hull, then there exists a monomorphism h, which must be an isomorphism. ( 1 ) o ( 5 ) : If in ( I ) , f is an injective hull and g an injective extension, then there exists a monomorphism h. Conversely, if in (I), f is a smallest injective extcnsion of A and g an injective hull, then there exists a monomorphism h, which is a section, hence an isomorphism.

LEMMA 4. Let V be a Grothendieck category and Q E V. Assume that Q has no proper essential extension. Then Q is injective. Proof. Let f : Q A be a monomorphism. Let 23 be the set of subobjects B of A with Q n I3 = 0. If {Bi}is a chain in 8,then (U Bi) n Q = U (Bin Q ) = 0 implies U BiE 23. By Zorn's lemma there exists a maximal object B' in 23. We shall show that Q -+A ---t A/B' is an isomorphism. Then f is a section and Q is injective by the assertion dual to Lemma 2. Because of Ker(gf) = B' n Q= 0, we get that gf is a monomorphism. Consider Q as a subobject of A and Q' = g ( Q ) as a subobject of AIB'. Let C C A/B' and Q' n C = 0. Then Q n g-'(C) C g-l(Q' n C ) = B' and Q ng - l ( C ) C Q, hence Q ng-'(C) C B' n Q = 0. On the other hand, g-'(C) 1B', hence we get g-'(C) = B' because of the maximality of B'. C = gg-'(C) = 0 because g is an epimorphism, that is, gf is an essential monomorphism. By hypothesis we get that gj is an isomorphism. --f

THEOREM 1. I f %? is a Grothendieck category with a generator then each object in %? has an injective hull. Proof. T o each object A E %? we shall construct a maximal essential extension. By Corollary 1, this will not have a proper essential extension any more, hence by Lemma 4 it will be injective. Into the class of all

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-

proper essential monomorphisms in %? we introduce an equivalence relation f g if and only if D( f ) = D ( g ) . Then by the strong axiom of choice we assign to each noninjective object B E 9? a proper essential extension. T o the injective objects, we assign the identical morphisms, since they do not possess a proper essential extension by Proposition 1. Now we construct a sequence of essential extensions

for all ordinals a. If the sequence has been constructed up to B, , let -+ B,+l be the essential extension determined above by the strong axiom of choice. If /3 is a limit, then we define B, = 1% B, for all a < p as in Corollary 1. Then for all a , the monomorphism A -+ B, is an essential monomorphism. Now we want to show that this sequence will become constant after a certain ordinal. Since for noninjective objects B, , the extensions must be proper by construction, the object B, , from where on the sequence will be constant, will be injective. Let G be a generator in ‘3and G‘ be an arbitrary subobject of G. Let a < /3 be ordinal numbers. We form the set (G’, a , p) of morphisms f :G‘+ B, for which there is a morphism g : G + B, such that the diagram

B,

G’ +G

is commutative. Hence, (G’, a, /I) C Homu;(G’,B,), For < p’, we have (G’, a, /3) C (G’, a, this sequence must become constant because Homu;(G’, B,) has only a set of subsets. Since G has also only a set of subobjects G’, there exists even an ordinal a* > 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

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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

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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

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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.

ABELIAN CATEGORIES

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

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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

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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

<

<

<

Y-Z

-

as the diagonal A 2, for by the dual assertion of Section 4.3, Lemma 3(c) with b also X -+ 2 is a monomorphism. Consequently, c is a monomorphism. Let f : A -+ B be a morphism in d.If a and a' are monomorphisms in A and if Z and 2' are the cofiber products off with a and a' respeca', then we get a commutative diagram tively, and if a

<

X

B

b'

* z

where b and 6' are monomorphisms, and 2 -+2' is uniquely determined by X + X ' and 6'. f defines a functor T ( f ) : T ( A )-+ T ( B ) which, with the notations of the diagram, assigns to an object a in T ( A ) the object b in T ( B ) ,such that a a' implies T(f )(a) = b b' = T(f )(a'). Since T ( A )and T ( B )are directed small categories, T (f ) is a functor. I f f : A+ B is a monomorphism in a? and if b, b' E T(B),then in the commutative diagram

<

<

4.

224

ABELIAN CATEGORIES

X

B-X’

b‘

bf and b’f are also monomorphisms and are objects of T(A). If b \< b’, then bf b’f. Thus we get a functor T+(f ) : T ( B )+ T ( A ) .If a E T ( A ) and b = T( f )(a), then the commutative diagram

<

fl

b

1

B-Y

implies that f as well as bf are monomorphisms, and that we have a bf. Thus for a monomorphism f wr: have

<

LEMMA1. T is a functor from A into the category of small categories with isomorphism classes of functors as morphisms. Proof, T h e definition of T implies trivially T(1,) lT(,), Let morphisms f : A -+ B and g : B -+ C in d be given. Let a E T(A). By Section 2.6, Lemma 3, we have T ( g ) T ( f ) ( a )g T ( g f ) ( a ) . Since all diagrams in T ( C ) must be commutative, this isomorphism is a natural isomorphism, T ( g ) T(f)s T(gf). Given an additive functor 9 : A? + V. We construct a functor 9,* : T ( A )+ V for each object A E d . If a E T ( A ) is given, then it defines an exact sequence O + A 4 X + Cok a + 0 Since S is not necessarily exact, 9 does not necessarily preserve the kernel A of X -+ Cok a, when applied to the above exact sequence. Let

4.13

225

FUNCTOR CATEGORIES

us define FA*(a) as the kernel of F ( X ) - + R ( C o k a ) , then we get a commutative diagram with an exact row

0

- - FA* ( u )

F(X)

F(Cok a)

where F ( A ).+ FA*(a) exists uniquely because ( S ( A )+F ( X )-+ F(C0k a)) = 0

If a

X

< a’

=

in T(A),then there exists a morphism x : X -+ Y in a?with R(a), Y = R(a’), and xu = a’. Therefore, we get a commutative

diagram with exact rows 0

0

-

-

FA*@)

F A *(a’)

-

‘a

9(X)

F(Cok a)

-

F(Y )

i,’

S(Cok a’)

where F ( C o k a) -+ F ( C o k a’) is determined by the natural morphism Cok a -+ Cok a’ and where F A * ( x ) exists because the right square of the diagram is commutative. Because of the uniqueness of the factorizations through kernels and cokernels, FA*(.) is uniquely determined by x. Now we have to show that the morphism S A * ( x ) does not depend on a’. T h u s let also y : X -+ Y with the choice of the representative for a ya = a’ be given. Then (x - y)a = 0. Hence x - y may be factored through Cok a. Then also 9 ( x - y) may be factored through F ( C o k a), and we have F ( x - y)iu = 0, hence F ( x ) i , = F ( y ) i a . T h e above diagram implies iu,FA*(x)= iu&”*(y) and F A * ( x ) = .FA*(y). T h u s FA*is defined on T ( A ) . T h e functor properties follow trivially from the functor properties of 9 and the uniqueness of the factorizations. Let f : A -+ B be a morphism in d.Then there are functors FB*T(f ) : T ( A )--t %? and FA*: T ( A )-+ V defined. We construct a .FA*-+FB*T(f). Let a : A ---f X be a natural transformation 9,:* monomorphism in d, hence an object in T ( A ) ,then T ( f ) ( a )= b : B -+ Y

<

4.

226

ABELIAN CATEGORIES

is a monomorphism into the cofiber product Y of X and B over A . We get a commutative diagram with exact rows

0 -A

1

-

0-B-

X-

Cok a --+

1

1

0

Y-Cokb-0

- - -1 1-

This diagram induces another commutative diagram with exact rows

0

0

FA*(a)

F(X)

.F(Cok a)

FB*(b) -+ F ( Y )

F(Cok b )

I

We denote the morphism F A * ( u )+ FB*(b) = F B * T ( f ) ( a ) by Ft*(a). I t is obviously uniquely determined by f and a. If g : B --t C is another morphism in d , then by the uniqueness the diagram

<

<

a’ in T ( A ) is given, then T ( f ) ( a )= b b’ = is commutative. If a T ( f ) ( a ’ ) .With the same argument as for the uniqueness of Sf*(a)one shows that the morphism %‘*(a) + FB*(b’) is uniquely determined. Therefore, the diagram

is commutative, where x is a representative for a \< a’ and y = T(f )(x) is a representative for b b’. Consequently, Ft* is a natural transformation. Now we assume that %‘ is a Grothendieck category. Then there exist By Section 2.5 there exists a morphism direct limits of the functors SA*.

<

lim T ( f ) : lim gB*T(f) lim FB* +

+

--f

+

4.13

FUNCTOR CATEGORIES

227

Furthermore, Sf*induces a morphism lim #F,*: lim FA* -+ lim P B * T ( f ) + --f +

T h e composition of these two morphisms will be denoted by (RS)( f) and 1 5 SA* with ( R g ) ( A ) .Then ( R T ) ( f ): (RF)(A)-+ ( R F ) ( B )is defined such that the diagram

is commutative. ( R F ) ( f ) is uniquely determined by the fact that all diagrams of this form are commutative for all a E T(A). T h e vertical arrows are the injections into the direct limit. 9 ( a ) = 19FA*(a) implies (RF)(l,) = I ( R s ) ( A ) . Fg*(b) Ft*(a)= F $ ( a ) i m p l L ( R S ) ( g ) ( R T )j) ( = ( R P ) ( g f ) .Hence, (RS)is a functor from d' to %?. T h e construction of FA*(.) defines a morphism F ( A )3 $'*(a) such that for all a \< a' the diagram S(A)

' FA*(a')

F A *(a)

is commutative. Thus we get a morphism 9 ( A )--t S A * ( u )-+ ( R S ) ( A ) , which is independent of the choice of a. Since for f : A ---t B, the diagram

m4

1

-.fw)

FA*(a)

is commutative, also

S(f)

S *(a)

1

*B*(b)

1 1 -

.F(A)

(RS)(A)

F(f)

(R.W(f)

F(B)

(Rm(B>

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.

<

LEMMA 4. Let .F : A? -+% be an additive functor which preserves monomorphisms. Then ( R F ) is left exact.

230

Proof. Let a

4. ABELIAN

CATEGORIES

< a’ in T ( A )be given. First, we show that the morphism

S A * ( x ) : F A * ( a )+ .FA*(a’) is a monomorphism. We form the cofiber

product A A X

Y-2

T h e composed morphism a” : A + 2 is a monomorphism because in the cofiber product the morphism X -+2 is a monomorphism. Thus we get a diagram

where the two inner quadrangles and the outer quadrangle are commutative, but not necessarily the right triangle. However, because of a a‘ a” the left triangle is commutative. Since by hypothesis S ( x ) is a monomorphism, we have S A * ( a ) ---+ S(2) and also S A * ( a )+ S A * ( a ’ )are monomorphisms. Let an exact sequence 0 + A 1,B 5 C --f 0 together with an object b E T ( B ) be given. Then we get a commutative diagram with exact rows

< <

f

O--+A+Bg’C+O

4 bl 4 bf

0-A-Y-2-0

where the right square is a cofiber product. T h e properties of the cofiber product imply that c is a monomorphism and Y --f 2 is an epimorphism. By construction, we have A _C Ker( Y + 2). T o show the converse, we consider the corresponding diagram with Y / Ainstead of 2. Then we get a morphism 2 -+ Y / A such that ( Y + Y / A )= ( Y + 2 + Y/A).This means that Ker( Y + 2 ) C A. Since 1, , 6 , and c are monomorphisms, we may complete this diagram by the 3 x 3 lemma. Let U = Cok(b)

4.13

23 1

FUNCTOR CATEGORIES

and V = Cok(c), then U g V because Cok(lA) = 0. If we apply 9to the diagram and form the corresponding kernels, then we get a commutative diagram with exact rows and columns 0

0

1

0

0

- - 1 -1-

_ +

Ker(d)

SB*(b)

%FA *(bf)

qY )

d

1 1 1

.Fc*(c)

S(Z)

1

.F(U ) I F(U )

(TA*(bf) 9( Y )-+ .F(U ) ) = 0 implies that there is exactly one -+

morphism T A * ( b f ) + .FB*(b) which makes the diagram commutative. -+ S B * ( b )-+ Sc*(c)) = 0, hence SA*(bf) +S B * ( b ) But then (SA*(bf) may be uniquely factored through Ker(d). Consequently, Ker(d)

=%*(bf

1.

For b

< b’, we get a commutative diagram with exact rows 0 0

_f

- -

%* T + ( f ) ( b )

1

gB*(b>

1

gA*T+(f)(b’)

__f

gB*(b’)

gC*

T(g)(b)

1

-

%c*T(g)(b’)

which after the application of the direct limit becomes the exact sequence lirnF,+

o + liin 6* ~ + (1 j) (zw)(B) +

lirnsg*

+

lim sc* T(g) --f

From the proof of Lemma 3, we know already that Ii$FA*T+(f) = ( R g ) ( A ) and 1% T,+= ( R g ) ( f ) . By definition ( R 9 ) ( g )= lim T ( g )1% Tg*. T o prove the assertion of the lemma, it is sufficient to show that & l m T ( g ) is a monomorphism. Since for c < c’ the morphism FC*(c) + S c * ( c ’ ) is a monomorphism, the morphisms F c * ( c ) + (RS)( C) are monomorphisms by Section 4.7, Lemma 3. By Section 4.7, Theorem l(b), 1 5 T ( g ) is a monomorphism.

LEMMA 5 . ( R 9 ) is an additivefunctor. Proof. Let A and B be objects in A and let S = A @ B be the direct sum. Let an object c E T ( S )be given. If we consider A by A -+S -% X

4.

232

ABELIAN CATEGORIES

as a subobject of X and correspondingly B C X , then the morphism X ---t X / A @ XIB induced by X -+ X / A and X -+ X / B is a monomorphism because the kernel is A n B = 0. T h e morphism (S X -+ X / A @ X / B ) = d is again a monomorphism and we have c d in T ( S ) . ( A -P S -+ XIA) = 0 and (B -+ S -+ X / B ) = 0 imply that d is the direct sum of the monomorphisms ( A -+ S+ X / B ) = a and (B -+ S -+ S / A ) = b. Hence the cokernel of d is the direct sum of the cokernels of a and b. Since 9 is an additive functor, 9 preserves the

> <

decompositions into direct sums. Since kernels preserve direct sums, the kernel of F ( X / A @ X / B )-+ 9 ( C o k d) is the direct sum of the kernels of S ( X / A )-+ .F(Cok b) and 9 ( X / B )-+ 9 ( C o k a). This construction preserves the corresponding injections and projections. Hence, FS*(d) = S A * ( a )@ SB*(b).The application of the direct limit gives ( R 9 ) ( S )= ( R 9 ) ( A )@ ( R S ) ( B ) .In fact it is sufficient to form the direct limit over those objects d E T ( S ) that may be written as a direct sum of objects a E T ( A ) with objects b E T ( B ) because, to each object c, there exists d. such an object d with c

<

LEMMA 6. R : %(at’, U ) -+ %(A?,‘4?)is u left exact functor. Proof. Let 0 -P 95 93 5 Z -P 0 be an exact sequence in %(at’, U). For each A E d and each a E T(A),we get a commutative diagram with exact rows and columns 0 0

0

0

0

0

-1- -1 - 1 -1 -1 - -I .FA*@)

B”*(U)

%*(a)

S(X)

B(X)

%(X)

S(Cok a)

B(Cok a )

%(Cok a)

The morphisms S A * ( a )--f gA*(a)constructed in this way are obviously natural transformations with respect to A ELZ? and a E T(A). Thus we may apply the direct limit over T ( A )in the first row to get an exact sequence ( R d ( A ) , ( R ~ ( A (RB)(A) ) ,( R Z ) ( A ) 0 (RS)(A)

-

where the morphisms are uniquely determined by OT and and are natural in A by construction. Because of the uniqueness it is clear that R is a

4.13

FUNCTOR CATEGORIES

233

functor. R is a left exact functor by the definition of the exactness in (zI(d, U). With these lemmas we now can solve easily the universal problem described in the beginning of this section.

THEOREM 1. Let d be a small abelian category and U be a Grothendieck category. Then 2 ( d , 9) is a rejexive subcategory of %(d, U ) . The rejector Ro : 21(&’, U ) -+Ll(&’, U ) is called the zeroth right-derived functor. Proof. We know that it is sufficient to solve the corresponding universal U ) , and a natural transformation problem. Let 9E a(&, U ) , 9 E I?(&, v : 9+ 9 be given. By two-fold application of Lemma 2, we get a commutative diagram

where p’ is the natural transformation which corresponds to ( R 9 ) and is constructed similarly to p. $ and $’ are uniquely determined by v. By Lemma 3 ( R F )preserves monomorphisms. By Lemma 4, ( R ( R 9 ) ) is left exact. T h u s the universal problem is solved. Furthermore,

R09

=

(R(RF)).

COROLLARY 1. Under the hypotheses of Theorem 1 , 2 ( d , U ) is an abelian category.

Proof. T h e direct sums in 2 ( d , U ) and N ( d , U ) coincide because the direct sum of left exact functors is left exact. Furthermore, the null functor is left exact. By the theorem of Section 4 . 1 , 2 ( d , U ) is an additive category. Let 9) : 9+ 9 be a natural transformation of left exact functors. T h e kernel of this morphism in F u n c t ( d , %?-hence, argumentwise formed-preserves kernels, that is, is left exact. We denote this functor by Ker(q). This functor has, also in 2 ( d ,U ) , the property of a kernel. Let A/ be the cokerncl of in F u n c t ( d , U ) . Let : 9 + 3? be a morphism in 2 ( d , V) with +y = 0. Then we get a commutative diagram

+

9-L9 23+ROY

234

4.

ABELIAN CATEGORIES

where $" is uniquely determined by 4. Hence ROT cokernel of cp in 2 ( d , W ) . In %(d, %?)we have an exact sequence

=

Cok(cp) is the

0 + Ker(7) + 9 % 3? --t 0 Since R is left exact by Lemma 6, Ro is also left exact. So we get the exact sequence 0 -+ R°Ker(T)-+ 9 + fix in %(d, U) (and also in 2 ( d , W ) ) because B = ROB, since 9 is left W ) . Then we get exact. Let 9 be the cokernel of Ker(cp) -+ 9 in %(d, an exact sequence 0 -+Ker(v) + 9+ R o 9 Since 9 is the coimage of cp in 2 1 ( d , %) and Ker(.r) is the image of v, we have that 9 g Ker(7). Hence we also have R o 9 = ROKer(7). T h e last two exact sequences show that R o 9 is the coimage of cp in 2 ( d ,U) and ROKer(7) is the image of cp in 2 ( d , %). Hence, 2 ( d ,%?)is an abelian category.

THEOREM 2. Let d be a small abelian category and Ab be the category of abelian groups. Then 2 ( d ,Ab) is a Grothendieck category with a generator. Proof. F u n c t ( d , Ab) has coproducts; coproducts of additive functors Ab) is cocomplete. Since f?(d, AB) is a full are additive, hence, %(d, Ab), , 2 ( d , Ab) is also cocomplete reflexive subcategory of % ( d (Section 2. I I , Theorem 3). We show that the Grothendieck condition holds in F u n c t ( d , Ab). Let {Si} be a directed family of subfunctors of B and X' be a subfunctor of 9.Since subfunctors are kernels in F u n c t ( d , Ab), the corresponding monomorphisms are pointwise monomorphisms. Since limits and colimits are formed pointwise in F u n c t ( d , Ab), intersections and unions of functors are formed pointwise also:

((u%) n

2 )(4=

((u6(4)n mj u P i ( 4 =

fl

%(A))

Thus direct limits in F u n c t ( d , Ab) are exact. Since they preserve additive functors, they are also exact (LI(d, Ab). Since kernels in 2 ( d , Ab) coincide with kernels in PI(&, Ab), the monomorphisms also coincide. Direct limits preserve monomorphisms in %(d, Ab), hence

4.13

235

FUNCTOR CATEGORIES

they also preserve monomorphisms in 2 ( d ,Ab), for direct limits of left exact functors are again left exact by the Grothendieck condition. Consequently, direct limits in 2 ( d ,Ab) are exact, that is, the Grothendieck condition holds. T o show that 2 ( d ,Ab) is locally small, it is sufficient to know that there is a generator in 2 ( d ,Ab). We claim MAE&hA = G is a generator. First, hA is left exact for all A E d.Then the coproduct of left exact functors is left exact. (Section 2.7, Corollary 2), hence G E 2 ( d ,Ab). Let q~ and I,L be two different natural transformations from 9 to 9 in 2 ( d ,Ab). Then there is at least one A ~d with q(A) # #(A).Hence 9 ( A ) to B(A) are different. the product morphisms from By the Yoneda lemma these are induced by the morphisms Morj(G, 9') and Mor,(G, I,L) because we have Mor,(G, 9)g 9 ( A ) and Mor,(G, 9)E 9(A). Since 9 ( A ) # 0 for all A E&', also Mor,(G, 9) # e, for all 9E 2 ( d , Ab). Consequently, G is a generator for 2 ( d , Ab).

nAEd

nAEd

nAEd

COROLLARY 2. Q(d, Ab) is an abelian category with an injective cogenerator. Proof.

T h e corollary is implied by Section 4.9, Corollary 3.

THEOREM 3. The contravariant representation functor h : d -+ 2 ( d ,Ab) is full, faithful, and exact. Proof. We denote the injective cogenerator of 2 ( d , Ab) by X . T h e functor Mor,( -, X ) : 2 ( d ,Ab) -+ Ab is faithful and exact by definition of X . By Section 4.3, Lemma 2 and Section 2.12, Lemma 1, a sequence in 2 ( d ,Ab) is exact if and only if the image under Mort(--, X ) is exact. Let 0 -+ A -+ B -+ C ---f 0 be an exact sequence in d . Then the sequence 0 -+ hC -+ hB -+ hA is exact, since for all D E d the sequence 0 -+ hc(D) + hB(D) hA(D) is exact and since kernels in Q ( d Ab) , are formed pointwise. Thus the sequences Morj(hA,X ) -+Mor,(hB, X ) -+ Mor,(hc, X ) -+ 0 and X ( A )-+ X ( B )-+ X ( C ) -+ 0 are exact. But X is a left exact functor, thus even 0 -+X ( A )-+ X ( B )-+ X ( C ) -+0 and also 0 -+ Morf(hA,X ) -+ Morf(hB,X ) -+ Mor,(hc, X ) -+ 0 are exact. Thus by the above remark 0 + hC-t h B +

hA

+0

is exact, We know already from Section 2.12, Proposition 2 that the representation functor is full and faithful.

236

4.

ABELIAN CATEGORIES

4.14 Embedding Theorems We have investigated the importance of full faithful functors in Section 2.12. For abelian categories, there is an additional very important notion, namely that of an exact functor. Again the behavior of functors with respect to diagrams is of interest. Since the corresponding diagram schemes, however, are not abelian categories in general, we shall have to reformulate the exactness. Let us discuss the example of a part of the assertion of the 3 x 3 lemma. Given a commutative diagram with exact columns and an exact first and second row in an abelian category V,

0

__f

0

0

0

1

1

1

1 -1 -1

B,

0 +c,

B,

B, 40

--c,

c 3

1

1

1

0

0

0

0

then the third row is also exact. How can we formulate this assertion in the language of diagram schemes ? First, let a diagram scheme 9 with the corresponding objects Ai’, Bi‘, Ci’ (i = I , 2, 3) and 0’ be given. When we define the morphisms of 9, we may already take into account the existing commutativity relations. Let 9be the functor which maps 9 to our given diagram. T h e assertions about the exactness have to be checked in V. But we can say in 9 for which pairs of morphisms we have to check the exactness, namely for

( 0-+ A;‘, A,’

-+

A,’)

,..., (B3’-+ C,’, C3‘+ 0’)

however, not for

(0’+ C,‘, Cl’

--f

C,’),

(C,’

+Ci,

C,’

-+

C,’)

and

(C,’

--f

C,’, C3’ + 0 )

If these pairs of morphisms become exact after the application of S, then by the 3 x 3 lemma also the pairs (O’+ C,’, C,’+ C2’),...,

4.14

EMBEDDING THEOREMS

237

(C,’ + C,’, C,‘-+ 0’) will become exact after the application of 9. But here we did not yet take into account that Y ( 0 ‘ )shall be the zero object of V. Certainly we can ask for this property separate from the exactness conditions. But there is an exactness condition which implies this condition automatically. If (0’ 5 0’, 0’ A 0’) becomes exact after the application of 9, then Y ( 0 ‘ )can only be a zero object of g.Later we shall express other conditions by exactness conditions. First, we want to formalize the considerations we made up to now. Let 3 be a diagram scheme. A set E of pairs of morphisms in 9 is called a set of exactness conditions if we have R(a) = D(b) for each pair ( a , 6) E E, that is, if the two morphisms in a pair may be composed in 9.Let 9 : 9 + %‘be a diagram over 9 in V . We say that 9satisfies the exactness conditions E if for each pair (a, b) E E the sequence

-

F ( D ( a ) ) gF(d

Ojr

(R(a))

F(R(b))

is exact. Let us denote by E, thc exactness conditions for the zero object and the exactness of the columns and the first and second row and by E, the exactness conditions for the exactness of the last row of the given diagram, then the 3 x 3 lemma may be formulated in the following way. Each diagram B which satisfies the exactness conditions E, also satisfies the exactness conditions E, . If 9satisfies a set of given exactness conditions, then it is possible that certain parts of the diagram become commutative where the commutativity in 9 was not given or not recognizable. T h e commutativity of diagrams may also be expressed by a set K of pairs of morphisms in 9 for which (a, 6) E K always implies D(a) = D(b) and R(a) = R(b). Such a set K is also called a set of commutativity conditions. We say that B satisfies the commutativity conditions K if for each pair (a, b) E K we have F ( a ) = Y ( b ) . An exact categorical statement in an abelian category $? with respect to the diagram scheme 9 with the exactness conditions E and E‘ and the commutativity conditions K and K‘ is an assertion of the following form: Each diagram 9over 9 in %?which satisfies the exactness conditions E and the commutativity conditions K satisfies also the exactness conditions E’ and the commutativity conditions K’. Since the identities and compositions of morphisms may already be formulated in 9 and are preserved by the functor 9some of the notions in an abelian category may bc defined by exactness and commutativity conditions. Since we are only interested in functors 9 which satisfy the given exactness and commutativity conditions, we can formulate the defining exactness and commutativity conditions in V for the particular notions independently of the diagram scheme 9.

4.

238

ABELIAN CATEGORIES

T h e following assertions hold in any abelian category %': = 0 if and only if A 5 A A A is ' exact. ( A -+ B) = 0 if and only if A -+ B -!+ I3 is exact. A -+ B is a monomorphism if and only if 0 -+ A -+ B is exact. (A -+ B) = Ker(B -+ C) if and only if 0 -+A -+ B C is exact. S = A @ B with projections and injections S + A , S -+ B, A + S, and B -+ S respectively if and only if A --t S B is exact, B + S -+ A 1

A

-

-

isexact,(A-S+A) =(AAA),and(B+S+B)= (B-B). C + S is the morphism into the direct sum induced by C --+ A and C - t B if and only if ( C - + S - + A ) = ( C - + A ) and ( C - + S + B ) = (C -+ B). The diagram P-A

B-C

is a fiber product if and only if 0 + P

-

A @ B -+ C is exact.

Beyond these examples there are many more notions which may be represented in a similar way. I n particular, finite limits and colimits together with their universal properties may be defined in this way.

LEMMA1 . Let 9 : A? -+ %? be a faithful exact functor between abelian categories. Assume that the exact categorical statement deJined by (3,E, K , E', K ' ) is true in %?, Then it is also true in A?.

-

Proof. We have to show that a diagram 9: 9 A? which satisfies the exactness conditions E and the commutativity conditions K also satisfies the exactness conditions E and the commutativity conditions K'. By hypothesis, 9 3 : 9 4%? satisfies the conditions E' and K'. I n fact, if S satisfies E and K , then 5 9 3 satisfies conditions E and K because B is exact. Since B is faithful, the conditions K' have to be satisfied already in A?. We only have to show that a sequence A + B -+ C in A? is exact if 9(A) -+ 9(B) -+ 9 ( C ) is exact in %'.In fact, then E' also holds in A?. Let A 5 B -% C in A? be not exact. T h e n ( A -+ B + C) # 0 or (Ker(g) -t B -+ Cok(f)) # 0. Since B is faithful and exact, B preserves kernels, cokernels, and nonzero morphisms. Hence ($(A) -t 9(B) Y(C)) # 0 or (Ker(g(g)) %(B) C o k ( g ( f ) ) )# 0. Hence, also $(A) -+ B(B) %(C) cannot be exact (Section 4.3, Lemma 1).

-

-

-

-+

4.14

EMBEDDING THEOREMS

239

With this lemma we can test the truth of an exact categorical statement via faithful exact functors. Since a diagram consists always only of a set of objects and morphisms, it is interesting to know if each diagram in an abelian category is already in a small abelian category. Later on we shall see that for small abelian categories there are faithful exact test functors into the category of abelian groups. PROPOSITION 1. Each set of objects in an abelian category lies in a small full exact abelian subcategory.

Proof. Let dobe the full subcategory of the abelian category V with the given set of objects in V as objects. Now we construct a sequence of full subcategories diof V by the following construction. If diis given, then let di+l consist of the kernels and cokernels of all morphisms of dias well as of all direct sums of objects of diwhere the kernels, cokernels, and direct sums have to be formed in V and where we take for each morphism only one kernel and cokernel and to each finite set of objects only one direct sum. Let di+l be the full subcategory of %? defined by these objects. Since diis small also di+l is small. Furthermore, we have diC if, for example, we use A as the kernel of 0 : A --t A. Thus dois in = diand 9is a small full exact abelian subcategory of V. By definition g is a small full subcategory. A3 contains the zero object of V as kernel of an identity and the morphism sets of A3 form abelian groups in the same way as they do in V. Furthermore, for each finite set of objects in A9 there exists a direct sum in 28 since the finite set has to lie already in one of the di. Therefore, g is an additive category. Furthermore, kernels and cokernels of morphisms in 93 coincide with kernels and cokernels in V by definition, and they exist. T h e natural morphism from the coimage into the image of a morphism in 37 coincides with the one formed in %',so it has an inverse morphism which is also in A3. Thus 9 is abelian and the embedding is exact.

THEOREM 1. Let d be a small abelian category. Then there exists a covariant faithful exact functor 9 : d -+ Ab from d into the category of abelian groups. Proof. We apply Section 4.13, Theorem 3 and Corollary 2. T h e contravariant representation functor h : d +2 ( d ,Ab) is faithful and exact. Let X be an injective cogenerator in Q(d, Ab). Then the contravariant representable functor hlorf(-, X ) : 2 ( d ,Ab) -+ Ab) is faithful and exact by the definition of the injective cogenerator. T h e composition of these two functors is covariant, faithful, and exact and we have

4.

240

ABELIAN CATEGORIES

Mor,(-, S ) h g .X. Hence, S : &' Ab is a covariant faithful exact functor. Now by Lemma 1 , it is sufficient to test the truth of exact categorical statements only in the category of abelian groups Ab. This is also true for an arbitrary abelian category V, since each diagram is already in a small abelian category d by Proposition I and since the exactness in &' and in V is the same. Since we can check the exactness and equality of morphisms in Ab elementwise, many proofs will be considerably simplified. We formulate this fact in the metatheorem that follows. ---f

METATHEOREM 1. A n exact categorical statement which is true in the category Ab of abelian groups is true in each abelian category.

As an application of this metatheorem, we show that in each abelian category the lattice of subobjects of an object is modular. A lattice is called modular if for elements A , B, and C of the lattice A C C implies A u ( B n C) = ( A u B ) n C. We always have A u ( B n C ) C ( A u B ) n C by the hypothesis A _C C. T o prove the equality in the lattice of the subobjects, we have to show that the morphism A u ( B n C) C ( A u B ) n C is an isomorphism. For the formulation of an intersection and a union, we may use finite limits and colimits. Hence, the modularity of the lattice of the subobjects of an object in an abelian category is an exact categorical assertion. We need check it in Ab only. But if c E ( A u B ) n C, then c = a b with a E A and b E B. Since A C C we get c - a = b E C, hence b E B n C. This proves that c = a b E A u ( R n C), that is, A u ( B n C) = ( A u B ) n C.

+

+

COROLLARY 1. The lattice of the subobjects of an object in an abelian category is modular. With our example of the 3 x 3 lemma we were only able to cover a part of the lemma as an exact categorical statement. Although in this case it is easy to prove the existence of the morphisms in the lower row, which make the diagram commutative, it is of principal interest to carry even this task over into another category by a suitable functor. This problem deals with two diagram schemes with the same objects where the morphisms of the first diagram scheme are also morphisms of the second diagram scheme, but in the second diagram scheme there are more morphisms. Let 9 be a diagram scheme with the exactness conditions E and the commutativity conditions K. Let 9' be another diagram scheme with the exactness conditions E' and the commutativity conditions K'. Let

4.14

EMBEDDING THEOREMS

24 1

.f : $3 -+ 9‘be a functor which is bijective on the objects. (Y, 9,9’, E, K , E‘, K‘) defines a full exact categorical statement with respect to an abelian category V of the following form: T o each diagram 9 : 9-+ V which satisfies the exactness conditions E and the commutativity conditions K , there exists a diagram 9’ : 9’ V with 9’9 = 9 which satisfies the exactness conditions E‘ and the commutativity conditions K‘. Hence the 3 x 3 lemma is a full exact categorical statement which is true -+

in each abelian category.

LEMMA 2. Let 9 : 9+ V be a full faithful exact functor between abelian categories. Let the full exact categorical statement defned b y (9, B,%,E, K , E’, K‘) be true in V . Then it is also true in a. Proof. Assume that 9: 9+9 satisfies the conditions E and K. T h e n also 99 : 9 -+ V satisfies the conditions E and K , because 9 is exact. Hence there is a diagram 9“ : 9’+ V which satisfies the conditions E’ and K’. By Section 1.15, Lemma 2 , s ” may be uniquely factored through ~3 with a diagram 9’ : 9’+9and 9“ = 99’. Since 9 is faithful and exact and since 9F’ satisfies the conditions E’ and K ’ , so does 9’. This has already been proved in Lemma 1. By Proposition 1 we may decide each full exact categorical statement already in a small abelian category, namely in the small full exact abelian subcategory which contains all objects of the diagram 9: 9--t V. This category certainly depends on the choice of the diagram 9. However, if we show that a full exact categorical statement in each small full exact abelian subcategory of V is true, then it is also true in %. For the following considerations, we still need another theorem.

THEOREM 2. Let % be a cocomplete abelian category with a projective generator P. Let d be a small full exact abelian subcategory of V . Then there exists a full faithful exact covariant functor 9 : d + Mod, from at‘ into a category of R-modules. Proof. T h e proof goes analogously to the proof of Section 4.11, Theorem 1. Since P is not finite, we shall not try find epimorphisms from coproducts of P with itself to the particular objects, but only epimorphisms from some projective generator. Since each coproduct of copies of I ) is again a projective generator we choose the number of factors large enough such that each object A of d may be reached by an epimorphism 1l P + A. This is possible because d is small. Let us call 1l P = P‘ and R = Homc(P’, P’). Since P ‘ is a projective generator, the functor Hom%(P’,-) : %+Mod, is faithful and exact. We still have to show that the restriction 9 of Homv(P’, -) to the subcategory d’is full. Then 9is full, faithful, and exact. Let 9 = Homv(P‘, -).

242

4.

ABELIAN CATEGORIES

Let f : 9 A +9 B be given for objects A, B E d.We have to find a morphism f ’ : A + B with Ff ’ = f . Let a : P’ -+ A and b :P’ + B be epimorphisms. Since the ring R is projective we get a commutative diagram Ker(S a )

-R

9a

.FA

T h e morphism R + R may be represented in the form Bg because Homu(P’, P’) g Hom,(R, R ) by the functor Homa(P’, -). Since B is exact, Ker(Ba) B(Ker(a)). Since B is faithful, (Ker(Ba) -+R + R + F B ) = 0 implies (Ker(a) + P ‘ -+P‘+ B) = 0. Thus in the diagram Ker(a)

-

P’

A

there exists exactly one morphism f’ which makes the square commutative. Hence, the upper diagram becomes commutative also if we replace f by Ff ’. But since B a is an epimorphism we get f = 9f ’.

THEOREM 3 (Mitchell). Let d be a small abelian category. Then there exists a covariant full faithful exact functor F : d + Mod, from d into a category of R-modules. 2 ( d ,Ab) is contravariant, full, faithful, and exact. Let ho be the corresponding functor from d into the category Q0(d, Ab) dual to 2 ( d ,Ab) which is cocomplete by Section 4.13 and has a projective generator. Then ho is covariant, full, faithful, and exact. Let 9 be the small full exact abelian subcategory of i ? O ( d , Ab) which is generated by h o ( d )by Proposition I . Then by Theorem 2, there exists a full faithful exact functor 9+ Mod, for a ring R. Hence also d +G? + Mod, is covariant, full, faithful, and exact. As in the case of the Metatheorem 1, Lemma 2 and Theorem 3 imply the following result.

Proof. The functor h : d

--f

METATHEOREM 2. A full exact categorical statement which is true in all module categories is true in each abelian category. Now with this theorem we can also decide about the existence of

4.14

EMBEDDING THEOREMS

243

morphisms in relatively simple categories, namely module categories where one can compute elementwise. So the 3 x 3 lemma need only be proved with these means in an arbitrary module category. This then implies that it holds in all abelian categories. T h e best-known application of this theorem is the existence of the connecting homomorphism.

COROLLARY 2. Let the diagram 0

0

0

-1-1 1 1 1 -1 1 1 -1

B,

B, -0

c,

c,

D,

D,

D,

0

0

0

B,

0 +c1

be commutative with exact rows and columns in an abelian category %. Then there is a morphism 6 : A, --t D , called the connecting homomorphism such that the sequence A,-+A3-+Dl-+D,

is exact.

Proof. T h e assertion of the corollary is a full exact categorical statement. So we need only check it in a module category. We define the following application A, 3 a3 ++ 6, ++ 6,

H c2 ++ c1 ++

d, E D,

Here the elements are in the modules with the corresponding subscripts. Let b, be chosen such that b, is mapped onto b, by B, -+ B, . Since c, is mapped onto 0 by C, --f C,, c, is already an element in Cl which we denote by c1 . T h e only ambiguity of this application is the choice of

4.

244

ABELIAN CATEGORIES

b, . This choice is unique up to a summand b,‘ E B, . But we have 6, b,‘ b c, c,’ t+ c1 c,’ t+ d, because (B, + C,+ 0,) = 0. Obviously this map is a homomorphism which we denote by 6 : A, -+ D,. If 6(a,) = 0, then there exists a b,’ E B, with b,‘ H c1 by B, + C, . Therefore, b, - b,’ ++0 by B , -+ C,. Hence, there exists exactly one a, with a,- b, - b,’. But then a, w a, by A, + A,, hence A, --t A, -+ D, is exact. Let d, E D, be given such that d, is mapped to 0 by D, -+ D, , then there exists a c1 with c1 I-+ d, and c1 is mapped to 0 by C,-+ C,+ D , . Hence there exists a b, with b, t,c, and c1 I+ c2 . Therefore, b, is mapped to 0 by B,-+B,-+ C,, that is, there exists an a, with a , t t b , and b, w b,. By definition of 6 we have 6(a,) = d, . Consequently, the sequence A, -+A, .% D, D, is exact.

+

+

+

-

Problems 4.1.

Show that Example 2 of Section 4.1 is not an abelian category.

4.2.

The sequence 0

4.3.

Let Q be an abelian category. If the following diagram in %?

-+

A

-+

B

-+

0 is exact if and only if

A

-+

B is an isomorphism.

O-A’-B’-C-O

1. -1. ,

0-

A’ P-!-

,

1.

B ’ L C‘-

0

is commutative and if both rows are short exact sequences, then: (1) if u and w are monomorphisms, then w is a monomorphism; (2) if u and w are epimorphisms, then v is an epimorphism. 4.4

Dualize Lemma 2 of Section 4.3. 1

9

4.5. Let 0 -+ A -+ B -+ C -+ 0 be an exact sequence. The following are equivalent: (1) f is a section. (2) g is a retraction. (3) B = A @ C and f : A -+ A 0 C is the injection with respect to A. 4.6. Show that the category of ordered abelian groups is not an abelian category. An abelian group G is ordered if G is an ordered set (Section 1.1, Example 2) such that a < b implies a x
+

+

245

PROBLEMS

4.7, Show that the assertion of Section 4.5, Lemma 1 holds without the assumption that CIA is simple. 4.8. Find an object of finite length with infinitely many different subobjects in a n abelian category. 4.9.

Show that RMod is a Grothendieck category for each unitary associative ring R.

4.10. In a module category RMod the union (in the categorical sense) of submodules of a module is the sum (in the module theoretical sense) of these submodules. 4.11. Prove the Steinitz exchange theorem for vector spaces by Section 4.10, Theorem 5. 4.12. Show that in Section 4.9, Corollary 4 the module RIM for a local ring R is not a cogenerator, so that in general RIM is not a cogenerator.

IJ

4.13. An additive functor between abelian categories is faithful if and only if it reflects exact sequences (Section 4.14, Lemma 1). 4.14. Does Section 4.10, Corollary 1 hold for arbitrary equationally defined algebras instead of K-modules ? 4.15. (a) Let M E RMod. Let {Gljic,be the set of large submodules of M and {Ej)jg, G, = E, for suitable subset K C J be the set of simple submodules of M. Then and is called sock of M. (b) M E RMod is called commpact if M is compact in RModn.M is cocompact if and only if the sock of M is large in M and finitely generated.

IJjEK

4.16. Let Y : Y: + L9 be an additive functor between abelian categories with a n 'preserves injective objects. exact right adjoint functor F.Then 9 4.17. Let V be an abelian cocomplete category with a finitely generated generator and let G, G' E V;. (a) If for all simple objects U there is a commutative diagram

G

f

*u

P(U) with an epimorphismf and a projective object P ( U ) , then G is a generator. (b) If for all simple objects U there is a commutative diagram

U

V

G'

4U ) with a monomorphism g and a projective object I ( U ) ,then G' is a cogenerator.

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Appendix Fundamentals of Set Theory We shall give an outline of those facts of set theory which are not too well known from naive set theory. As a basis we shall use the set of axioms of Godel and Bernays. T h e difference between sets and classes and the consequences of the strong axiom of choice play an important part in the theory of categories. Since the axiomatic description of set theory is very formalistic, we shall try to express most of the formulas in ordinary language. Observe that, for an axiomatic representation of a theory, we prove theorems on and within this theory, but that the models which satisfy the axioms of this theory do not belong to the theory itself. So axiomatic set theory involves computations with the given formulas; the “class of all sets,” however, or better, a model for the class of all sets will not be given. T h e axioms and the theorems derived from the axioms, however, should always have a meaning for naive set theory. We agree on the following symbols: class variables X , Y , 2,...; special classes 0 , U, A , , A, ,...; set variables x, y , z ,..., and formulas y ~ ,$,... . We may use subscripts with the symbols so that we have countably many symbols at hand in this way. Equality = and element of E are used between set variables, class variables, and special classes, where on both sides different kinds of these symbols may be used. Logical symbols are: “not” l, “or” v, “and” A , “implies” =-, “if and only if” e, “there exists” V, “there exists exactly one” V!, and “for all” A . T h e symbol precedes formulas. T h e symbols v , A , =-,o are used between formulas. T h e symbols V, V!, and A are used in front of variables, they are put in parentheses, and are followed by a formula or some other sequence of symbols. There are relations between the logical symbols through which all logical symbols may be reduced to the three logical symbols 1, A , and V (and the equality sign). T h e other symbols may be considered as abbreviations in the following way: v

t,h

is equivalent to

7(1~ 241

A

+).

248

APPENDIX

p)

*+

p)

o

is equivalent is equivalent (A X)p) is equivalent ( V ! X)p) is equivalent

+

to +p) A +). to (p * +) A (+ p). to -,((V X) p)). to ( V X)p) A (( V Y)p) 3 ( X = Y ) ) .

A formula is inductively defined for variables or special classes A,

r by

(1) A E r is a formula. (2) If p) and are formulas, then also

+

lv, cp A +, and ( V x)p) are formulas where x may be replaced by any other set variable and where we admit abbreviations (e.g., with other logical symbols). (3) Only those sequences of symbols which arise from (1) and (2) are called formulas.

If one of the variables occurs together with one of the so-called quantifiers V, A , or V ! (e.g., ( V X)..., (A X )..., ( V ! y )...), then the variable is called a bound variable, otherwise it is called a free variable. The axioms of set theory are subdivided into several groups. T h e axioms of group A are: x is a

(All

class

Each set is a class. X

E

Y => Xis a set

(A21

Each class which is an element of another class is a set. (A U ) ( U E X O UYE)

X

=

Y

(A31

Axiom of extensionality: If two classes have the same elements, then they are equal. (If two classes are equal, then they also have the same elements by the logical properties of the equality sign.) (A U, w)(V W ) ( X E w o x

=u

v x = W)

(A4)

For any two sets u, v there exists another set w which contains exactly u and v as elements. Only sets may occur as elements of classes or sets. In particular elements are not objects different from sets, contrary to the view of naive set theory. Talking about elements is nothing more than the colloquial transcription of the symbol E. We introduce a number of abbreviations which will be admitted also

FUNDAMENTALS OF SET THEORY

249

in formulas except for the first two abbreviations. Here “:=” has the meaning of “is an abbreviation of”.

X is a set. X is a class but not a set. 1 X = Y . 1XE Y . the set defined by (A4) which contains exactly x and y and which is uniquely determined by (A3). {xx}, hence {x} = {xx}. {{x}{xy}}, the ordered pair of x and y. X.

XCY xcY

.= ..-.-

(x1(x2 --.x,)) for all positive integers n. Thus finite ordered sets are defined. (( V u)(u E X)), X is empty. ((V u)(u E X A u E Y)), X and Y do not have a common element. (A u, v, w ) ( ( ( v u )E X A ( w u ) E X ) => w = v), the subclass of X that contains only ordered pairs contains to each u, at most one pair (vu), that is, X has uniquely defined values. ( A u)(u E X * u E Y ) .

(xcy )

A

( X # y).

T h e axioms of the other groups B, C, and D are:

There is a class A which contains the ordered pair (xy) if and only if x E y holds. (A A, B)(V C)(A u)((u E A

A

u E B ) o u E C)

(B2)

For any two classes A and B there exists a class C, the intersection A n B of A and B, which consists of exactly those sets which are elements as well of A as of B. (A A)(V B)(A u)(u 4 A o u E B )

(B3)

T o each class A there exists a class B, the complement -A of A, which contains exactly those sets which are not contained in A.

250

APPENDIX

T o each class A there exists a class B, the domain D(A) of A , which contains exactly the second components of the ordered pairs in A. (A A)(V B)(A Xy)((YX) E B -3 x E A )

(B5)

T o each class A there exists a class B, which contains an ordered pair if and only if the second component of the ordered pair is an element of A. Nothing is said about the elements of B which are not ordered pairs. (B5) serves to construct the product. (A A)(V B)(A xY)(<xY)E A

0

(YX) E B)

(B6)

T o each class A there exists a class B which contains as ordered pairs exactly the ordered pairs of A with reversed order.

*(yzx)

B)

(B7)

(A A)(V B)(h*,Y* z ) ( ( x y z ) E A -3 ( x z y ) E B )

(B8)

(A A)(V B)(A x , y , z ) ( ( x r z >E A

T o each class A there exists a class B which contains as triples exactly the triples of A where the order is changed in correspondence with (B7) or (B8). (V

41EmPtY(4 A (A x)(x E 0

-

(VY)(Y E a

A

x CY)))

(C1)

There exists a set a which has at least countably (injinitely) many elements. (A x ) ( V y ) ( l \ u, v)(u E o A o E x => u E Y ) (C2)

To each set x there exists a set y which contains the union of those sets which are elements of x. (A x)(Vy)(A u ) ( u C x 3

UEY)

(C3)

T o each set x there exists a set y which contains each subset of x as an element. (Ax,A)(Un(A) ~ ( V y ) ( A u ) ( u ~ y o ( V v ) ( uo o~) x~ ~A ) ) ) (C4)

To each class A with unique values (an application) and to each set x there exists a set y which consists of exactly those elements which are the values of the elements of x under the application A. Empty(A)

=>

( V u)(u E A

A

Ex(u, A ) )

(D)

25 1

FUNDAMENTALS OF SET THEORY

Axiom of foundation: Each nonempty class contains an element which is disjoint to the given class.

LEMMA1. There exists exactly one class 0 with (A u)(u .$ exactly one class ZL with (A U ) ( U E U ) .

a)

and

Proof. By axiom (Bl) there exists a class A. By (B3) the complement B = -A of A exists. By (B2) the intersection C of A and B with (A u)(u E C o u E A A u E B), that is, ( A u)(u E C o u E A A ( u E A)), exists. Hence we get ( A u)(u $ C) because u E A A (u E A ) is always wrong. We set C = 0.By (A3) the class 0 is uniquely determined. Let U be the complement of 0 (B3). Also U is uniquely determined. We call U the universal class and on that 0 is a set.

ca

the empty set. We shall show later

METATHEOREM OF CLASS FORMATION. Let cp(xl

no other free variables than x1 ,..., x , such that the following holds:

x,) be a formula with

. Then there exists exactly one class A

(A u)(u E A o ( V x1 ,..., x,)(u

=

(xl

.**

x,)

h

v(xl

.**

x,)))

Proof. ( I ) We may assume that in cp there is no special class at the left side o f E because of (A,EI')O((VX)(X= A k

A XEI'))

(2) We may assume that, except special classes and variables, there occur only E, 1, A , and V (with parentheses) in the formulas (and no equality sign) because of ( A =: I')o ((A

X)(X E

A

ox E

r))

(3) Let cp = ( x r E xs). If r = s, then cp = ( x , E X , ) . But x, E x, and E x ( x r , {x,}) a contradiction to axiom (D). We set x, E {x,} implies

A,

If r

= D.

If r

> s, then

< s, then we get by (Bl)

we get by (Bl) and (B6)

252

APPENDIX

By (B5), (B6), (B7), and (B8) we get in all three cases with 1 1<s
(4) Let v = (x,, E Ak). Then we get (B5)-(BS) we get for 1 Y n

< <

( V A,)(h XI ,**., x,)((x,

< I < n and

(v Ak)(XrE A, o X,

..*x,)

E

A, -3x,

E

E Ak). By

Ad

( 5 ) Now we make an induction with respect to the number of logical symbols l, A , and V. T h e necessary induction steps are

i v : By(B3) (A x1 I . . . , x,)((x,

* (A

(Vx):

*..

x,)

x1 ,..., x,)((x~

4% *.. x,)>

E A,

... x,)

E

-A, o 1q ( ~ 1 x,)) a * *

By(B4)

(6) We define A x B by

Furthermore, let An = A x An-l. I n particular, we get (A

U)(U

E U" o ( V x1 ,..., x,)(u = (xl

x,)))

We replace the class B(= -A,, A, n A,, D(A,)) defined by ( 5 ) by the class A = B n Un. Then by (A3) the class A is uniquely determined by (A

U)(U E

A o ( V x1 ,...,x,)(u

=

(xl

*.*

x,)

A

p(xl

x,)))

The class A constructed in the Metatheorem of class formation is also

253

FUNDAMENTALS OF SET THEORY

written as A = {(x, viations:

A-B AuB U X (IX cp(X) X-1 %(X) F((X))

... x,) 1 ~ ( x ,

xn)}. T h u s we get further abbre-

:= {x I X E A A x $ B}, := {x I X E A v x E B } , :r

{XI

(VY)(XEYA Y E X ) } ,

x

:= {x I (A Y)(Y E * x E Y ) h := {x I x XI, .= .- {x I ( V Y , .I(. = ( Y Z ) A (ZY) E X ) } , := B(X-l) (range of X ) := %(F n (U x X ) ) (image of X under F, that is, the class of those elements which occur as images of elements of X under the application of F).

c

A series of new formulas is defined by Rel(X) Equ.Rel(X)

:= XCU2,

.= .-

Rel(X) A (A x)(x E a ( X ) * (xx) A ( A X,Y)((XY)E X => ( 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


is a complete system of representatives. T h e axiom of choice implies in particular Zorn’s Lemma. We define a chain K in an ordered set X (in the sense of Section 1.1, Example 2) to be a subset of X such that for any two elements x , y E K always x y ory x holds. An upper bound for a chain K in X is an element b ( K ) E X such that x b ( K ) for all x E K . A maximal element m E X is an element with the property that m x implies m = x for all x E X.Observe that a chain may be empty and that every element of X is an upper bound for the empty chain.

<

<

<

<

ZORN’SLEMMA.If X is an ordered set and ;f each chain K in X has an upper bound, then there is a maximal element in X . We have to refer the reader to text books on set theory for the proof of this and the following lemma on ordinals.

LEMMA 2. Let K be a well-ordered set of ordinals 01, let y be the first ordinal with I K I < I y I and I a 1 < I y I for all 01 E K. Then there is an ordinal p with /3 < y and a: < p for all a E K.

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BIBLIOGRAPHY Brinkmann, H. B., and Puppe, D., “Kategorien und Funktoren,” Lecture Notes 18. Springer, Berlin, 1966. Eckman, B. (ed.), “Seminar on Triples and Categorical Homology Theory,” Lecture Notes 80. Springer, Berlin, 1969. Ehresmann, C., “CatCgories et structures.” Dunod, Paris, 1965. Eilenberg, S., et al. (eds.), “Proceedings of the Conference on Categorical AlgebraLa Jolla-1965.” Springer, Berlin, 1966. Freyd, P., “Abelian Categories.” Harper, New York, 1964. Gabriel, P., Des cattgories abeliennes. Bull. Soc. Math. France, 90, 323-448 (1962). Hasse, M., and Michler, L., “Theorie der Kategorien.” Deut. Verlag. Wiss., Berlin, 1966. Lambek, J., “Completion of Categories,” Lecture Notes 24. Springer, Berlin, 1966. 71, 40-106 (1965). MacLane, S., Categorical algebra, Bull. Amer. Math. SOC., Mitchell, B., “Theory of Categories.” Academic Press, New York, 1965.

257

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Index A

Alternative algebra, 147 Alternative ring, 147 Amalgamated product, 83 Analytic group, 155 Anticommutative ring, 146 Antiisomorphism, 19 Artinian category, 21 Artinian object, 21 Artinian power set, 21 Artinian ring, 217 Associative ring, 4, 146 Associativity, 2 Automorphism, 17 Axiom of choice, 18, 20, 200, 254 Azumaya, 190, 193, 202, 208

Abelian category, 158. 163, 164 Abelian group, 4 divisible, 14, 18, 148, 163 finite, 156 ordered, 244 torsionfree, 74, 156 Additive category, 158, 159 Additive functor, 178, 179, 222 Adjoint functor, 51, 91 theorem, 105, 110, 113 Adjunction of unit, 148 Affine algebraic group, 155 Algebra, 68, 126, 127, 147, 149 alternative, 147 canonical, 126 commutative, 33, 148, 156 equationally defined, 127 exterior, 148 finitely generated, 135, 138 free, 68, 130, 135 homomorphism, 126 surjective, 142 Jordan, 147, 149 Lie, 147, 149 nil, 148 nilpotent, 148 polynomial, 149, 156 generalized, 156 sub-, 130, 138 symmetric, 148, 156 tensor, 148, 156 universal, I20 Algebraic category, 126 Algebraic functor, 137, 139, 145 Algebraic theory, 120, 121, 136, 145 consistent, 121, I36 of groups, 125, 126, 155

B Balanced category, 18, 38, 112 Base ring extension, 149 Beck, 72 Bernays, 2, 247 Bifunctor, 39 Bimodule, 147 Bound variable, 248 C

Cancellable morphism, left, 14 right, 14 Canonical algebra, 126 Cartesian square, 82 CO-, 83 Categorical statement exact, 237, 240 full exact, 241, 242, Category, I, 2, 48 abelian, 158, 163, 164 of abelian groups, 4, 8, 9, 18, 21, 22, 24, 259

260

INDEX

28, 33, 38,49, 74, 88, 112, 116, 129, 145, 163, 239, 240 additive, 158, 159 algebraic, 126 of algebras, 64,147 artinian, 21 balanced, 18, 38, 112 cocomplete, 78, 80 with (monomorphic) coimages, 35 with colimits 78 of commutative C-algebras, 33 complete, 78, 85 with (finite) coproducts, 33 coreflexive sub-, 74 diagram, 24, 89 with difference cokernels, 28 with difference kernels, 27 directed, 88 directly connected, 58 discrete, 6, 81 dual, 12, 19 empty, 3 equivalent, 18, 55 of finite groups, 16 finitely cocomplete, 78, 90 finitely complete, 78, 85, 89 filtered, 87 functor-, 10, 221 Grothendieck, 181, 182, 188 ofgroups, 3, 8, 16, 18,21, 22, 24,28, 33, 38, 50, 74, 88, 129, 146 hausdorff topological spaces, 15, 49, 74, 88, 115 with (epimorphic) images, 35 with (finite) intersections, 34, 170 inversely connected, 60 isomorphic, 18 large diagram, 79 with limits, 78 locally cosmall, 22 locally noetherian, 206 locally small, 21, 112 of R-modules, 4, 8, 18, 28, 33, 38, 56, 88, 129, 147, 164, 203, 204, 210 of monoids, 55, 56 morphism, 25 noetherian, 21 pointed, 23 of pointed sets, 3, 8, 22, 28, 33, 38, 112, 129

of pointed topological spaces, 5,8,22, 24, 28, 33, 38, 112 of pointed topological spaces modulo homotopy, 5, 155 product, 39 with (finite) products, 31 reflexive, sub-, 73, 114 of rings, 4, 8, 9, 15, 17, 21, 22, 24, 28, 33, 38, 49, 56, 74, 115, 129, 146, 155, 156 ofsets, 3, 8, 13, 14, 17, 18, 20, 21,22. 23, 26, 28, 32, 33, 38, 49, 50, 55, 85, 88, 99, 101, 112, 129, 156 small, 6, 8, 10, 24 of small categories, 8 sub-, 9 of topological spaces, 5, 8, 15,21, 22, 24, 28, 33, 38, 49, 74, 85, 88, 112, 114, 115, 155 of topological spaces modulo homotopy, 5 with (finite) unions, 34, 170 universally directly connected, 59 with zero morphisms, 23 Center, 215, 216 Chain, 21, 255 maximal condition, 21 minimal condition, 21 of subobjects, 21, 181 Class, 2, 247 power, 21 special, 247 universal, 251 variable, 247 Coarsest topology, 112 Coirreducible object, 208 Commutative algebra, 33, 148, 156 Commutative diagram, 9, 24 Commutative group, 4, 155 Commutative polynomial algebra, 149 Commutative ring, 74, 115, 146 Commutativity condition, 237 Commutator factor group, 148 Compact hausdorfF space, 74, 114 Compact module, 245 CO-, 245 Compact object, 204 Compactification, Stone-Cech, 114 Complement, 249 Complete category, 78, 85

26 1

INDEX CO-, 78, 90 finitely, 78, 85, 89 Complex, 167 Composition, 2 Composition series, 174 factor of, 174 length of, 174 Condition commutativity, 237 exactness, 237 Grothendieck, 181 Congruence, 141 Connected category directly, 58 inversely, 60 universally directly, 59 universally inversely, 60 Connecting homomorphism, 243 Connection, 58 Consistent algebraic theory, 121, 136 Constant functor, 77, 79 Construction, see Standard construction Continuous functor, 86 CO-,86 Continuous map, 5 dense, 15 pointed, 5 Contractible pair, 69 Contravariant functor, 7 Correspondence of sets, 6 Counterimage, 34, 35, 97 Covariant functor, 7 Creation of difference cokernels, 70 of isomorphisms, 69 properties, 69 Cube lemma, 15

D Decomposable object, 190 in-, 190 Decomposition, Q-S-, 118 Dense continuous map, 15 Derived functor, zeroth right-, 233 Diagonal, 33, 159 CO-,33, 159 Diagram, 9, 24 category, 24, 89 commutative, 9, 24

empty, 86 large, 79 large diagram category, 79 scheme, 6, 24, 237 Difference cokernel, 26, 28, 70, 81, 163 kernel, 26, 81-86, 163 quotient object, 28 subobject, 28 Direct limit, 81, 88 exact, 185 Direct sum, 159 Directed category, 88 Directed family of subobjects, 182 Directly connected category, 58 universally, 59 Discrete category, 6, 81 Discrete topological space, 8, 112 Divisible abelian group, 14, 18, 148, 163 Domain, 3, 250 Double dual space, 12 Dual category, 12, 19 Dual standard construction, 62 Dual theorem, 13 Duality, 12 principle, 13

E Ehrbar, 118 Eilenberg, 62, 256 Element, nilpotent, 192 Embedding theorem, 236 Empty category, 3 Empty diagram, 86 Empty product, 31 Empty set, 251 Endofunctor, 62 Endomorphism, 17 of identity functor, 216 ring, 163 Epimorphic image, 34 Epimorphism, 14 relative, I18 relatively split, 141 Equationally defined algebra, 127 Equivalence, 18, 55 Equivalence relation, 6, 99 monomorphic, 99 pre-, 101 Equivalent category, 18, 55

262

INDEX

Equivalent monomorphism, 20 Essential extension, 197, 198 largest, 198 maximal, 198 Evaluation functor, 43, 45, 46 Evaluation map, 11 Exact categorical statement, 237, 240 full, 241, 242 Exact direct limit, 185 Exact functor, 179 Exact sequence, 166 short, 166 Exactness condition, 237 Exchange theorem, 209, 245 Extension base ring, 149 essential, 197, 198 injective, 198 largest essential, 198 maximal essential, 198 minimal injective, 198 smalles injective, 198 Exterior algebra, 148

F Factor of composition series, 174 Faithful functor, 44, 115, 116 Family, 253 Fiber product, 82-84, 168 CO-,83, 85, 169 Fiber sum, 83 Field, 148 skew-, 219 Filtered category, 87 Filtered colimit, 88 Filtered limit, 88 Final object, 22, 84, 86 Finite poduct, 84 Finite projective object, 210 Finitely cocornplete category, 78, 90 Finitely complete category, 78, 85, 89 Finitely generated algebra, 135, 138 Finitely generated module, 205 Finitely generated object, 204 Forgetful functor, 8, 129 Formula, 248 Free algebra, 68, 130, 135 Free algebraic theory, 123 Free object, relatively, 141

Free product, 33 Free variable, 248 Full exact categorical statement, 241, 242 Full faithful functor, 115, 117 Full functor, 44, 115, 116 Full matrix ring, 219 Functor, 6, 7 additive, 178, 179, 222 adjoint, 51, 91 adjoint functor theorem, 105, 110, 113 algebraic, 137, 139, I45 bi-, 39 category, 10, 221 cocontinuous, 86 constant, 77, 79 continuous, 86 contravariant, 7 covariant, 7 endo-, 62 evaluation, 43, 45, 46 exact, 179 faithful, 44, 115, 116 forgetful, 8, 129 full, 44, 115, 116 full faithful, 115, 117 halfexact, 180 image of a, 24 isomorphic, 18 leftexact, 180, 222 monadic, 68, 139, 140 multi-, 39 power set, 50 product, 55 projection, 39 representable, 10, 11, 14, 40, 47, 105 representable functor theorem, 109 representable sub-, 105 representation, 44, 116, 235 rightexact, 180 G

Gabriel, 2, 158 Generator, 110, 111, 141, 202, 245 CO-, 110, 112, 203, 213, 245 pro-, 211 set of, 110 Godel, 2, 247, 254, 255 Grothendieck, 2, 158 category, 181, 182, 188 condition, 181

263

INDEX Group, 3, 6, 125, 126, 127, 146 abelian, 4 affine algebraic, 155 algebraic theory of, 125, 126, 155 algebraic theory of commutative, 155 analytic, 155 category of finite, 16 commutator factor, 148 divisible abelian, 14, 18, 148, 163 finite abelian, 156 homomorphism, 4 homotopy, 155 ordered abelian, 244 quasi-, 145 semi-, 145 sub-, 21, 50 topological, 155 torsion, 74 torsionfree abelian, 74, 156 Groupoid, 101, 104

H H-object, 145 Half exact functor, 180 HausdorfT topological space, 8, 15, 18, 74, 114 compact, 74, 114 normal, 74, 114 Hilton, 120 Holder, 174, 176 Homomorphism algebra-, 126 connecting, 243 group-, 4 insertion, 156 module-, 4 theorem, 142, 164 Homotopy, 5 group, 155 Hull, injective, 195, 198 1

Ideal, nilpotent left, 217 Idempotent, 190 Identity, 2, 10 of n-th order, 125 Image, 34, 165, 253 CO-, 34, 35, 165

counter-, 34, 35, 97 epimorphic, 34 of functor, 24 monomorphic co-, 35 Indecomposable object, 190 Induced topology, 49 Inductive limit, 81 I n h u m , 81 Initial object, 22 Injection, 33, 78 Injective cogenerator, 203, 213 Injective extension, 198 minimal, 198 smallest, 198 Injective hull, 195, 198 Injective map, 14 ,253 Injective object, 195 relatively, 141 Insertion homomorphism, 156 Integral domain, 115, 148 Intersection, 33, 34, 97, 171, 249 co-, 34 Inverse limit, 81, 88 Inversely connected category, 60 Isomorphic categories, 18 Isomorphic functors, 18 Isomorphic morphisms, 17, 25 Isomorphic objects, 17 Isomorphism, 14, 17 anti-, 19 creation of, 69 natural, 18 theorems, 144, 172, 173

J Jordan algebra, 147, 149 module, 147 ring, 147 Jordan-Holder theorem, 174, 176

K Kan, 51, 108 Kernel, 28, 163 CO-, 28, 163 difference, 26, 81-86, 163 difference co-, 26, 28, 70, 81, 163 pair, 86, 141 Kleisli, 62, 136 Krull, 190, 193, 202, 208

264

INDEX

L Large diagram, 79 category, 79 Large subobject, 197 Largest essential extension, 198 Lattice modular, 240 of quotient objects, 170 of subobjects, 170 Lawvere, 2, 120 Left adjoint functor, 51, 91 Left exact functor, 180, 222 Left ideal, nilpotent, 217 Left root, 81 Left zero morphism, 22 Length of composition series, 174 object of finite, 177 of object, 177 Lie algebra, 147, 149 module, 147 ring, 146 Limit, 51, 77, 81, 89, 91, 97 CO-, 77, 78, 81, 91 direct, 81, 88 exact direct, 185 filtered, 88 filtered co-, 88 inductive, 81 inverse, 81, 88 projective, 81 Linton, 120 Local ring, 190, 202 Locally arcwise connected space, 74 Locally connected space, 74 Locally cosmall category, 22 Locally noetherian category, 206 Locally small category, 21, 112 Loop, 146

M M-object, 145 Map, 253 continuous, 5 dense continuous, 15 injective, 14, 253 order preserving, 3 pointed, 3 pointed continuous, 5

surjective, 14, 253 Matlis, 208 Matrix of homomorphisms, 162 Matrix ring, full, 219 Maximal condition for chains, 21 Maximal essential extension 198 Maximal subobject 21 Metatheorem, 240, 242 of class formation, 251 Minimal condition for chains, 21 Minimal injective extension, 198 Minimal subobject, 21 Mitchell, 242 Modular lattice, 240 Module, 4, 147 bi-, 147 cocompact, 245 compact, 245 finitely generated, 205 homomorphism, 4 Jordan, 147 Lie, 147 relatively projective, 196 Monad, 61, 62 CO-, 62 Monadic functor, 68, 139, 140 Monoid, 55, 56, 62, 145 ring, 56, 149 Monomorphic coimage, 35 Monomorphic equivalence relation, 99 Monomorphism, 14, 87 equivalent, 20 relatively split, 141 Moore, 62 Morita, 158, 210, 213, 217 Morphism, 1, 150 of algebraic theories, 121 antiiso-, 19 auto-, 17 category, 25 of diagrams, 25 endo-, 17 endomorphism ring, 163 epi-, 14 iso-, 14, 17 isomorphic, 17 matrix of, 161 mono-, 14, 87 zero, 22, 23 Multifunctor, 39

265

INDEX

N n-ary operation, 125 Natural isomorphism, 18 Natural numbers, 6 Natural transformation, 6, 9 Nilalgebra, 148 Nilpotent algebra, 148 Nilpotent element, 192 Nilpotent ideal, 217 Nine lemma, see Three-by-three lemma Noetherian category, 21 locally, 206 Noetherian object, 21, 206 Noetherian power set, 21 Noetherian ring, 206 Nonunit, 190 Normal hausdorff space, 74, 114 Normal quotient object, 28 Normal subgroup, 50 Normal suboboject, 28 Numbers, natural, 6 0 Object, 1 artinian, 21 cogroup, 151 coirreducible, 208 compact, 204 decomposable, 190 difference quotient, 28 difference sub-, 28 final, 22, 86 of finite length, 177 finite projective, 210 finitely generated, 204 Hopf, 145 indecomposable, 190 initial, 22 injective, 195 isomorphic, 17 large sub-, 197 lattice of sub-, 170 maximal sub-, 21 minimal sub-, 21 multiplicative, 145 noetherian, 21, 206 normal quotient, 28 normal sub-, 28

order of sub-, 21 projective, 141, 195 quotient, 20, 21 relatively free, 141 relatively injective, 141 relatively projective, 141 representing, 11, 47 simple, 174 sub-, 20 transfinitely generated, 205 zero, 22 Order identity of n-th, 125 of subobjects, 21 Order preserving map, 3 Ordered abelian group, 244 Ordered set, 3, 6 pre-, 81 Open set, 5 Operation, n-ary, 125

P Pair of adjoint functors, 51 Pointed category, 23 Pointed continuous map, 5 Pointed map, 3 Pointed set, 3 Pointed topological space, 5 Polynomial algebra, 149, 156 generalized, 156 Power class, 21 co-, 21, 22 Power set, 21 artinian, 21 functor, 50 noetherian, 21 Preequivalence relation, 101 Preordered set, 81 Preservation property, 69 Problem, see universal problem Product, 29, 81, 85, 86, 158, 250 amalgamated, 83 category, 39 CO-, 29, 33, 81, 83, 159 cofiber-, 83 empty, 31 fiber, 82-85 finite, 84 free, 31

INDEX

functor, 55 tensor, 33, 56, 152 Progenerator, 21 1 R-S-, 214 Projection, 31, 78 functor, 39 Projective limit, 81 Projective module, relatively, 196 Projective object, 141, 195 finite, 210 relatively, 141 Pullback, 82 Pushout, 83

Q Quantifier, 248 Quasigroup, 145 Quotient object, 20, 21 difference, 28 lattice of, 170 normal, 28 Quotient topology, 49

R R-S-progenerator, 214 Radical ring, 146 Range, 3, 253 Reduced representative, 132 Reflector, 74 co-, 74 Reflexion property 69 Reflexive subcategory, 73,114 co- 74 Regular space, 74 Relative epimorphism, 118 Relatively free object, 141 Relatively injective object, 141 Relatively projective module, 196 Relatively projective object, 141 Relatively split epimorphism, 141 Relatively split monomorphism, 141 Remak, 190, 193, 202, 208 Representable functor, 10, 11, 14, 40, 47, 105 theorem, 109 Representable subfunctor, 105 Representation functor, 44, 116, 235 Representative, reduced, 132 Representing object, 11, 47

Retraction, 29 Right adjoint functor, 51, 91 Right derived functor, zeroth, 233 Right exact functor, 179 Right root, 81 Right zero morphism, 22, 23 Ring, 4, 146 alternative, 147 anticommutative, 146 artinian, 217 associative, 4, 146 center of, 215, 216 commutative, 74, 115, 146 endomorphism, 163 full matrix, 219 homomorphism, 4 Jordan, 147 Lie, 146 local, 190, 202 monoid, 56, 149 noetherian, 206 radical, 146 semisimple, 21 7 simple, 21 7 unitary, 146 Root left, 81 right, 81 S

Skeleton, 19 Schmidt, 190, 193, 202, 208 Schur, 220, 221 Section, 29 Semigroup, 145 Semisimple ring, 217 Sequence exact, 166 short exact, 166 Set 2, 247 artinian power, 21 correspondence of, 6 empty, 251 of generators, 110 noetherian power, 21 open, 5 ordered, 3, 6 pointed, 3 power, 21

INDEX

preordered, 81 underlying, 129 variable, 247 Short exact sequence, 166 Simple object, 174 Simple ring, 217 semi, 217 Skew-field, 219 Small category, 6, 8, 10, 24 locally, 21, 112 locally co-, 22 Socle, 245 Solution set, 106 universal, 58, 60 Sphere, 155 Split epimorphism, relatively, 141 Split monomorphism, relatively, 141 Square Cartesian, 82 cocartesian, 83 Standard construction, 62 dual, 62 Statement exact categorical, 237, 240 full exact categorical, 214, 242 Stone-Cech compactification, I14 Subalgebra, 130, 138 Subcategory, 9 coreflexive, 74 reflexive, 73, 114 Subfunctor, representable, 105 Subgroup, 21, 50 Subobject, 20 chain of, 21, 181 difference, 28 directed family of, 182 large, 197 lattice of, 170 maximal, 21 minimal, 21 normal, 28 order of, 21 Subspace, topological, 21, 49 Sum direct, 159 fiber-, 83 Supremum, 81 Surjective algebra homomorphism, 142 Surjective map, 14, 253 Symmetric algebra, 148, 156

267 T

Ti-space, 74 Tensor algebra, 148, 156 Tensor product, 33, 56 of algebraic theories, 152 Theorem of Beck, 72 dual, 13 exchange, 209, 245 of Krull-Remak-Schmidt-Azumaya, 190, 193, 202, 208 of Matlis, 208 meta-, 240, 242 metatheorem of class formation, 251 of Mitchell, 242 of Morita, 210, 213, 217 Theory algebraic, 120, 121, 136, 145 of groups, 125, 126, 155 free, 123 consistent, 121, 136 morphism of, 121 tensor product of, 152 Three-by-three (3 x 3) lemma, 172, 236 Topological group, 155 Topological space, 5 compact, 74, 114 discrete, 8, 112 hausdorff, 8, 15, 18, 74, 114 locally arcwise connected, 74 locally connected, 74 normal hausdorff, 74, 114 pointed, 5 regular, 74 totally disconnected, 74 Topological subspace, 21, 49 Topology, coarsest, 87 Torsionfree abelian group, 74, 156 Torsiongroup, 74 Totally disconnected space, 74 Transfinitely generated object, 205 Transformation, natural, 6, 9 Tripel, 62 CO-,62 Tychonoff, 114

U Underlying set, 129 Union, 33, 34, 170 co-, 34

268

INDEX

Unit, 48, 190 non-, 190 Unitary associative ring, 4 Unitary module, 4, 147 Unitary ring, 146 homomorphism, 4 Univers, 2 Universal algebra, 220 Universal class, 251 Universal enveloping algebra, 149 Universal problem, 27, 31, 56, 58, 59 Universal solution, 58, 60 Universally directly connected category, 59 Universally inversely connected category, 60 Upper bound, 255 Urysohn, 114

v Variable bound, 248

class, 241 free, 248 set, 247 Vector space, 4

W Watts, 204 Well ordered set, 255

Y Yoneda lemma, 41, 42,46

Z Zero morphism, 22, 23 Zero object, 22, 23, 158 Zeroth right derived functor, 233 Zorn, 199, 203, 209, 255 Zurich, 120

Pure and Applied Mathematics A Series of Monographs and Textbooks Editors

Paul A. Smith and Samuel Eiienberg Columbia University, New York

1 : ARKOLD SOMMEHFELD. Partial Differential Equations in Physics. 1949 (Lectures

on Theoretical Physics, Volume VI) 2: REINHOLD BAER.Linear Algebra and Projective Geometry. 1952 BUSEMANN AN11 PAUL KELLY.Projective Geometry and Projective 3: HERBERT Metrics. 1953 4: STEFAN BERGMAN A N D M. SCHIFFER. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. 1953 5 : RALPHPHILIPBOAS,JR. Entire Functions. 1954 BUSEMANX. The Geometry of Geodesics. 1955 6: HERBERT CHEVALLEY. Fundamental Concepts of Algebra. 1956 7 : CLAUDE 8: SZE-TSENHu. Homotopy Theory. 1959 Solution of Equations and Systems of Equations. Second 9: A. M. OSTROWSKI. Edition. 1966 10: J. D I E U D O N N Treatise ~. on Analysis. Volume I, Foundations of Modern Analysis, enlarged and corrected printing, 1969. Volume 11, 1970. Curvature and Homology. 1962. 11: S. I. GOLDBERG. HELGASON. Differential Geometry and Symmetric Spaces. 1962 12 : SIGURDCR Introduction to the Theory of Integration. 1963. 13: T. H. HILDEBRANDT. Local Analytic Geometry. 1964 14 : SHREERALI ABIIYANKAR. L. BISHOPA X D RICHARD J. CRITTENDEN. Geometry of Manifolds. 1964 15: RICHARD A. GAAL.Point Set Topology. 1964 16: STEVEN Theory of Categories. 1965 17: BARRYMITCHELL. P. MOI~SE. A Theory of Sets. 1965 18: ANTHONY

Pure end Applied Mathematics A Series of Monographs and Textbooks

19: GTJSTAVECHOQUET. Topology. 1966 20: Z. I. BOREVICH A N D I. R. SHAFAREVICH. Number Theory. 1966 21 : Josh LUISMASSERA A N D J U A N JORCE SCHAFFER. Linear Differential Equations and Function Spaces. 1966 22 : RICHARD D. SCHAFER. An Introduction to Nonassociative Algebras. 1966 23: MARTINEICHLER.Introduction to the Theory of Algebraic Numbers and Functions. 1966 24 : SHREERAM ABHYANKAR. Resolution of Singularities of Embedded Algebraic Surfaces. 1966 25 : FRANCOIS TREVES. Topological Vector Spaces, Distributions, and Kernels. 1967 D. LAXand RALPHS. PHILLIPS. Scattering Theory. 1967 26: PETER 27: OYSTEINORE.The Four Color Problem. 1967 28 : MAURICE HEINS.Complex Function Theory. 1968 29 : R. M. BLUMENTHAL A N D R. K. GETOOR. Markov Processes and Potential Theory. 1968 30: L. J. MORDELL. Diophantine Equations. 1969 31 : J. BARKLEY ROSSER.Simplified Independence Proofs : Boolean Valued Models of Set Theory. 1969 32: WILLIAMF. DONOCHUE, JR. Distributions and Fourier Transforms. 1969 33: MARSTON MORSEAND ST~WART S. CAIRNS.Critical Point Theory in Global Analysis and Differential Topology. 1969 34: EDWINWEISS.Cohomology of Groups. 1969 35 : HANSFREUDENTHAL A N D H. DE VRIES.Linear Lie Groups. 1969 36: LASZLO FUCHS. Infinite Abelian Groups : Volume I. 1970 37: KEIO NACAMI.Dimension Theory. 1970 38: PETER L. DUREN. Theory of HP Spaces. 1970 39 : BODOPAREICIS. Categories and Functors. 1970

I n prejaran’on EDUARD PRUCOVEEKI. Quantum Mechanics in Hilbert Space.