CATEGORIES AND FUNCTORS This is Volume 39 in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAUL A. SMITH AND SAMUEL EILENBERG A complete list of titles in this series appears at the end of this volume CATEGORIES AND FUNCTORS Bodo Pareigis UNIVERSITY OF MUNICH MUNICH, GERM AN Y 1970 AC ADE MIC PRESS New York • London This is the only authorized English translation of Kategorien und Funktoren - Eine Einführung (a volume in the series "Mathematische Leitfäden,'* edited by Professor G. Rothe), orig- inally in German by Verlag B. G. Teubner, Stuttgart. 1969 COPYRIGHT © 1970, 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. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W1X 6BA LIBRARY OF CONGRESS CATALOG CARD NUMBER: 76-117631 PRINTED IN THE UNITED STATES OF AMERICA Contents Preface vii 1 • Prcliminary Notions 1.1 Definition of a Category 1 1.2 Functors and Natural Transformations 6 1.3 Representable Functors 10 1.4 Duality 12 1.5 Monomorphisms, Epimorphisms, and Isomorphisms .... 14 1.6 Subobjects and Quotient Objects 20 1.7 Zero Objects and Zero Morphisms 22 1.8 Diagrams 24 1.9 DifTerence Kernels and DifTerence Cokernels 26 1.10 Sections and Retractions 29 1.11 Products and Coproducts 29 1.12 Intersections and Unions 33 1.13 Images, Coimages, and Counterimages 34 1.14 Multifunctors 39 1.15 The Yoneda Lemma 41 1.16 Categories as Classes 48 Problems 49 2* Adjoint Functors and Limits 2.1 Adjoint Functors 51 2.2 Universal Problems 56 2.3 Monads 61 2.4 Reflexive Subcategories 73 2.5 Limits and Colimits 77 2.6 Special Limits and Colimits 81 2.7 Diagram Categories 89 v vi CONTENTS 2.8 Constructions with Limits 97 2.9 The Adjoint Functor Theorem 105 2.10 Generators and Cogenerators 110 2.11 Special Cases of the Adjoint Functor Theorem 113 2.12 Füll and Faithful Functors 115 Problems 118 3* Universal Algebra 3.1 Algebraic Theories 120 3.2 Algebraic Categories 126 3.3 FreeAlgebras 130 3.4 Algebraic Functors 137 3.5 Examples of Algebraic Theories and Functors 145 3.6 Algebras in Arbitrary Categories 149 Problems 156 4* Abelian Categories 4.1 Additive Categories 158 4.2 Abelian Categories 163 4.3 Exact Sequences 166 4.4 Isomorphism Theorems 172 4.5 The Jordan-Hölder Theorem 174 4.6 Additive Functors 178 4.7 Grothendieck Categories 181 4.8 The Krull-Remak-Schmidt-Azumaya Theorem 190 4.9 Injective and Projective Objects and Hulls 195 4.10 Finitely Generated Objects 204 4.11 Module Categories 210 4.12 Semisimple and Simple Rings 217 4.13 Functor Categories 221 4.14 Embedding Theorems 236 Problems 244 Appendix. Fundamentals of Set Theory 247 Bibliography 257 Index 259 Thinking—is it a social function or one of the brains ? Stanislaw Jerzy Lee Preface In 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. Düring the last decades, however, interest proeeeded 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. The theory of categories and functors abstracts the coneepts "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 strueture. 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 "simi- larities" in exaet terms. Parallel to this fact there is a unification. Thus 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 familiär 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. The 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 fleld. 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 em- bedding 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 Gödel 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. 1 Preliminary Notions The first sections of this chapter introduce the preliminary notions of category, functor, and natural transformation. The 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. The 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. The verifica- tion 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. hl Definition of a Category In addition to mathematical objects modern mathematics investigates more and more the admissible maps defined between them. One familiär 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 dement 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 ^ be a class of objects A, B, C,... e Ob ^ together with (1) A family of mutually disjoint sets {Mor^(Ay B)} for all objects Ay Bs ff, whose elements /, g> A,... G MO%(^4, B) are called morphisms and 1 2 1. PRELIMINARY NOTIONS (2) a family of maps {Mor^, B) x Mor^S, C)3 (f,g)^gfe MorrfA, C)} for all A, By C e Ob ^, called cotnposttions. & is called a category if ^ fulfills the following axioms: (1) Associativity: For all ^4, £, C, Z> e Ob ^ and all fe Mor^, B), g G Mor^(5, C), and h e Mory(C, ß) we have %/) = (W (2) Identity: For each object E Ob ^ there is a morphism 1A e Mor^^i, A), called the identity, such that we have f\A =f and lAg =g for all B, C e Ob <T and all /e Mor^, 5) and ^ e Mor^(C, Therefore the class of objects, and the class of morphism sets, as well as the composition of morphisms, always belong to a category The compositions have not yet been discussed in our example of sets, whereas the morphisms correspond to the discussed maps.