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Jiˇr´ı Adámek Horst Herrlich George E. Strecker Abstract and Concrete Categories The Joy of Cats Dedicated to Bernhard Banaschewski The newest edition of the file of the present book can be downloaded from http://katmat.math.uni-bremen.de/acc The authors are grateful for any improvements, corrections, and remarks, and can be reached at the addresses Jiˇr´ı Adámek, email: [email protected] Horst Herrlich, email: [email protected] George E. Strecker, email: [email protected] All corrections will be awarded, besides eternal gratefulness, with a piece of delicious cake! You can claim your cake at the KatMAT Seminar, University of Bremen, at any Tuesday (during terms). Copyright (cid:13)c 2004 Jiˇr´ı Ada´mek, Horst Herrlich, and George E. Strecker. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. See p. 512 ff. 2 PREFACE to the ONLINE EDITION Abstract and Concrete Categories was published by John Wiley and Sons, Inc, in 1990, and after several reprints, the book has been sold out and unavailable for several years. Wenowpresentanimprovedandcorrectedversionasanopenaccessfile. Thiswasmade possible due to the return of copyright to the authors, and due to many hours of hard work and the exceptional skill of Christoph Schubert, to whom we wish to express our profound gratitude. The illustrations of Edward Gorey are unfortunately missing in the current version (for copyright reasons), but fortunately additional original illustrations byMarcelErné,towhomadditionalspecialthanksoftheauthorsbelong,counterbalance the loss. Open access includes the right of any reader to copy, store or distribute the book or parts of it freely. (See the GNU Free Documentation License at the end of the text.) Besides the acknowledgements appearing at the end of the original preface (below), we wish to thank all those who have helped to eliminate mistakes that survived the first printing of the text, particularly H. Bargenda, J. Ju¨rjens W. Meyer, L. Schröder, A. M. Torkabud, and O. Wyler. January 12, 2004 J. A., H. H., and G. E. S. 3 PREFACE Scienceshaveanaturaltendencytowarddiversificationandspecialization. Inparticular, contemporary mathematics consists of many different branches and is intimately related to various other fields. Each of these branches and fields is growing rapidly and is itself diversifying. Fortunately, however, there is a considerable amount of common ground — similar ideas, concepts, and constructions. These provide a basis for a general theory of structures. The purpose of this book is to present the fundamental concepts and results of such a theory, expressed in the language of category theory — hence, as a particular branch of mathematics itself. It is designed to be used both as a textbook for beginners and as a reference source. Furthermore, it is aimed toward those interested in a general theory of structures, whether they be students or researchers, and also toward those interested in using such a general theory to help with organization and clarification within a special field. The only formal prerequisite for the reader is an elementary knowledge of set theory. However, an additional acquaintance with some algebra, topology, or computer science will be helpful, since concepts and results in the text are illustrated by various examples from these fields. One of the primary distinguishing features of the book is its emphasis on concrete categories. Recent developments in category theory have shown this approach to be particularly useful. Whereas most terminology relating to abstract categories has been standardized for some time, a large number of concepts concerning concrete categories has been developed more recently. One of the purposes of the book is to provide a reference that may help to achieve standardized terminology in this realm. Another feature that distinguishes the text is the systematic treatment of factorization structures, which gives a new unifying perspective to many earlier concepts and results and summarizes recent developments not yet published in other books. The text is organized and written in a “pedagogical style”, rather than in a highly economical one. Thus, in order to make the flow of topics self-motivating, new concepts are introduced gradually, by moving from special cases to the more general ones, rather than in the opposite direction. For example, • equalizers (§7) and products (§10) precede limits (§11), • factorizations are introduced first for single morphisms (§14), then for sources (§15), and finally for functor-structured sources (§17), • the important concept of adjoints (§18) comes as a common culmination of three separate paths: 1. via the notions of reflections (§4 and §16) and of free objects (§8), 2. via limits (§11), and 3. via factorization structures for functors (§17). Each categorical notion is accompanied by many examples — for motivation as well as clarification. Detailed verifications for examples are usually left to the reader as implied exercises. It is not expected that every example will be familiar to or have relevance 4 for each reader. Thus, it is recommended that examples that are unfamiliar should be skipped, especially on the first reading. Furthermore, we encourage those who are working through the text to carry along their favorite category and to keep in mind a “global exercise” of determining how each new concept specializes in that particular setting. The exercises that appear at the end of each section have been designed both as an aid in understanding the material, e.g., by demonstrating that certain hypotheses are needed in various results, and as a vehicle to extend the theory in different directions. They vary widely in their difficulty. Those of greater difficulty are typically embellished with an asterisk (∗). The book is organized into seven chapters that represent natural “clusters” of topics, and it is intended that these be covered sequentially. The first five chapters contain the basic theory, and the last two contain more recent research results in the realm of concrete categories, cartesian closed categories, and quasitopoi. To facilitate references, each chapter is divided into sections that are numbered sequentially throughout the book, and all items within a given section are numbered sequentially throughout it. We use the symbol to indicate either the end of a proof or that there is a proof that is sufficiently straightforward that it is left as an exercise for the reader. The symbol D means that a proof of the dual result has already been given. Symbols such as A 4.19 are used to indicate that no proof is given, since a proof can be obtained by analogy to the one referenced (i.e., to item 19 in Section 4). Two tables of symbols appear at the end of the text. One contains a list (in alphabetical order) of the abbreviated names for special categories that are dealt with in the text. The other contains a list (in order of appearance in the text) of special mathematical symbols that are used. The bibliography contains only books and monographs. However, each section of the text ends with a (chronologically ordered) list of suggestions for further reading. These lists are designed to aid those readers with a particular interest in a given section to “strike out on their own” and they often contain material that can be used to solve the more difficult exercises. They are intended as merely a sampling, and (in view of the vast literature) there has been no attempt to make them complete1 or to provide detailed historical notes. Acknowledgements We are grateful for financial support from each of our “home universities” and from the Natural Sciences and Engineering Research Council of Canada, the National Academies of Sciences of Czechoslovakia and the United States, the U.S. National Science Foun- dation, and the U.S. Office of Naval Research. We particularly appreciate that such support made it possible for us to meet on several occasions to work together on the manuscript. Our special thanks go to Marcel Erné for several original illustrations that have been incorporated in the text and to Volker Ku¨hn for his efforts on a frontispiece that the 1Indeed, although some could serve as a suggested reading for more than one section, none appears in more than one. 5 Publisher decided not to use. We also express our special thanks to Reta McDermott for her expert typesetting, to Ju¨rgen Koslowski for his valuable TEXnical assistance, and to Y. Liu for assistance in typesetting diagrams. We were also assisted by D. Bressler and Y. Liu in compiling the index and by G. Feldmann in transferring electronic files betweenManhattanandBremen. WeareespeciallygratefultoJ.Koslowskiforcarefully analyzing the entire manuscript, to P. Vopˇenka for fruitful discussions concerning the mathematical foundations, and to M. Erné, H.L. Bentley, D. Bressler, H. Andréka, I. Nemeti, I. Sain, J. Kincaid, and B. Schröder, each of whom has read parts of earlier versions of the manuscript, has made suggestions for improvements, and has helped to eliminate mistakes. Naturally, none of the remaining mistakes can be attributed to any of those mentioned above, nor can such be blamed on any single author — it is always the fault of the other two. srohtua eht 6 Contents Preface to the Online Edition 3 Preface 4 0 Introduction 9 1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 I Categories, Functors, and Natural Transformations 19 3 Categories and functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5 Concrete categories and concrete functors . . . . . . . . . . . . . . . . . . 60 6 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 II Objects and Morphisms 97 7 Objects and morphisms in abstract categories . . . . . . . . . . . . . . . . 99 8 Objects and morphisms in concrete categories . . . . . . . . . . . . . . . . 130 9 Injective objects and essential embeddings . . . . . . . . . . . . . . . . . . 149 III Sources and Sinks 165 10 Sources and sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 11 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 12 Completeness and cocompleteness . . . . . . . . . . . . . . . . . . . . . . 208 13 Functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 IV Factorization Structures 233 14 Factorization structures for morphisms . . . . . . . . . . . . . . . . . . . . 235 15 Factorization structures for sources . . . . . . . . . . . . . . . . . . . . . . 253 16 E-reflective subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 17 Factorization structures for functors . . . . . . . . . . . . . . . . . . . . . 286 V Adjoints and Monads 299 18 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 19 Adjoint situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 20 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 7 VI Topological and Algebraic Categories 351 21 Topological categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 22 Topological structure theorems . . . . . . . . . . . . . . . . . . . . . . . . 376 23 Algebraic categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 24 Algebraic structure theorems . . . . . . . . . . . . . . . . . . . . . . . . . 401 25 Topologically algebraic categories . . . . . . . . . . . . . . . . . . . . . . . 410 26 Topologically algebraic structure theorems . . . . . . . . . . . . . . . . . . 422 VII Cartesian Closedness and Partial Morphisms 429 27 Cartesian closed categories. . . . . . . . . . . . . . . . . . . . . . . . . . . 431 28 Partial morphisms, quasitopoi, and topological universes . . . . . . . . . . 445 Bibliography 463 Tables 467 Functors and morphisms: Preservation properties . . . . . . . . . . . . . . . . . 467 Functors and morphisms: Reflection properties . . . . . . . . . . . . . . . . . . 467 Functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Functors and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Stability properties of special epimorphisms . . . . . . . . . . . . . . . . . . . . 468 Table of Categories 469 Table of Symbols 474 Index 478 GNU Free Documentation License 512 18th January 2005 Chapter 0 INTRODUCTION There’s a tiresome young man in Bay Shore. When his fiancée cried, ‘I adore The beautiful sea’, He replied, ‘I agree, It’s pretty, but what is it for?’ Morris Bishop 18th January 2005 9