Categories and functors

Cover ......Page 1
Title Page ......Page 4
Contents ......Page 6
Preface ......Page 8
1.1 Definition of a Category ......Page 10
1.2 Functors and Natural Transformations ......Page 15
1.3 Representable Functors ......Page 19
1.4 Duality ......Page 21
1.5 Monomorphisms, Epimorphisms, and Isomorphisms ......Page 23
1.6 Subobjects and Quotient Objects ......Page 29
1.7 Zero Objects and Zero Morphisms ......Page 31
1.8 Diagrams ......Page 33
1.9 Difference Kernels and Difference Cokernels ......Page 35
1.11 Products and Coproducts ......Page 38
1.12 Intersections and Unions ......Page 42
1.13 Images, Coimages, and Counterimages ......Page 43
1.14 Multifunctors ......Page 48
1.15 The Yoneda Lemma ......Page 50
1.16 Categories as Classes ......Page 57
Problems ......Page 58
2.1 Adjoint Functors ......Page 60
2.2 Universal Problems ......Page 65
2.3 Monads ......Page 70
2.4 Reflexive Subcategories ......Page 82
2.5 Limits and Colimits ......Page 86
2.6 Special Limits and Colimits ......Page 90
2.7 Diagram Categories ......Page 98
2.8 Constructions with Limits ......Page 106
2.9 The Adjoint Functor Theorem ......Page 114
2.10 Generators and Cogenerators ......Page 119
2.11 Special Cases of the Adjoint Functor Theorem ......Page 122
2.12 Full and Faithful Functors ......Page 124
Problems ......Page 127
3.1 Algebraic Theories ......Page 129
3.2 Algebraic Categories ......Page 135
3.3 Free Algebras ......Page 139
3.4 Algebraic Functors ......Page 146
3.5 Examples of Algebraic Theories and Functors ......Page 154
3.6 Algebras in Arbitrary Categories ......Page 158
Problems ......Page 165
4.1 Additive Categories ......Page 167
4.2 Abelian Categories ......Page 172
4.3 Exact Sequences ......Page 175
4.4 Isomorphism Theorems ......Page 181
4.5 The Jordan-Hölder Theorem ......Page 183
4.6 Additive Functors ......Page 187
4.7 Grothendieck Categories ......Page 190
4.8 The Krull-Remak-Schmidt-Azumaya Theorem ......Page 199
4.9 Injective and Projective Objects and Hulls ......Page 204
4.10 Finitely Generated Objects ......Page 213
4.11 Module Categories ......Page 219
4.12 Semisimple and Simple Rings ......Page 226
4.13 Functor Categories ......Page 230
4.14 Embedding Theorems ......Page 245
Problems ......Page 253
Appendix: Fundamentals of Set Theory ......Page 256
Bibliography ......Page 266
Index ......Page 268