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Sigma Series in Pure Mathematics -- Volume 1

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Horst Herrlich, George E. Strecker

Category Theory. Third Edition


xvi+402 pages, hard cover, ISBN 978-3-88538-001-6, EUR 48.00, 2007

This is a by now classical text in mathematics. It gives an introduction to category theory assuming only minimal knowledge in set theory, algebra or topology. The book is designed for use during the early stages of graduate study -- or for ambitious undergraduates. Each chapter contains numerous exercises for further study and control.

The attempt is made to present category theory mainly as a convenient language -- one which ties together widespread notions, which puts many existing results in their proper perspective, and which provides a means for appreciation of the unity that exists in modern mathematics, despite the increasing tendencies toward fragmentation and specialization.

The fact that the book appears in a 3rd edition proves that the authors achieved their goals.

The more advanced book "Abstract and Concrete Categories", which the authors of this book wrote together with Jiri Adamek, is also available from Heldermann Verlag as a free electronic publication.



Contents:

Preface ix
Preface to the second edition xi
Preface to the third edition xii
     
  I. Introduction 1
     
  II. Foundations  
1 Sets, classes, and conglomerates 9
     
  III. Categories  
2 Concrete categories 13
3 Abstract categories 15
4 New categories from old 23
     
  IV. Special Morphisms and Special Objects  
5 Sections, retractions, and isomorphisms 32
6 Monomorphisms, epimorphisms, and bimorphisms 38
7 Initial, terminal, and zero objects 46
8 Constant morphisms, zero morphisms, and pointed categories 48
     
  V. Functors and Natural Transformations  
9 Functors 53
10 Hom-functors 61
11 Categories of categories 64
12 Properties of functors 67
13 Natural transformations and natural isomorphisms 77
14 Isomorphisms and equivalences of categories 86
15 Functor categories 93
     
  VI. Limits in Categories  
16 Equalizers and coequalizers 100
17 Intersections and factorizations 107
18 Products and coproducts 115
19 Sources and sinks 126
20 Limits and colimits 133
21 Pullbacks and pushouts 138
22 Inverse and direct limits 151
23 Complete categories 155
24 Functors that preserve and reflect limits 166
25 Limits in functor categories 171
     
  VII. Adjoint Situations  
26 Universal maps 177
27 Adjoint functors 194
28 Existence of adjoints 207
     
  VIII. Set-Valued Functors  
29 Hom-functors 217
30 Representable functors 221
31 Free objects 231
32 Algebraic categories and algebraic functors 236
     
  IX. Subobjects, Quotient Objects, and Factorizations  
33 (E, M)-Categories 249
34 (Epi, extremal mono) and (extremal epi, mono) categories 255
35 (Generating, extremal mono) and (extremal generating, mono) factorizations 267
     
  X. Reflective Subcategories  
36 General reflective subcategories 275
37 Characterization and generation of E-reflective subcategories 281
38 Algebraic subcategories 288
     
  XI. Pointed Categories  
39 Normal and exact categories 294
40 Additive categories 305
41 Abelian categories 318
     
Appendix: Foundations 328
Bibliography 332
Index of Symbols 381
Index 383



Horst Herrlich is professor of mathematics at the University of Bremen, Germany: Homepage

Email: horst.herrlich@t-online.de
George E. Strecker is professor of mathematics at the Kansas State University at Manhattan, U.S.A.: Homepage

Email: strecker@math.ksu.edu