Theory of Categories (eBook)

Front Cover 1
Theory of Categories, Volume 17 6
Copyright Page 7
Contents 12
Preface 8
Chapter I. Preliminaries 18
Introduction 18
1. Definition 18
2. The Nonobjective Approach 19
3. Examples 20
4. Duality 21
5. Special Morphisms 21
6. Equalizers 24
7. Pullbacks, Pushouts 26
8. Intersections 27
9. Unions 28
10. Images 29
11. Inverse Images 30
12. Zero Objects 31
13. Kernels 31
14. Normality 33
15. Exact Categories 35
16. The 9 Lemma 37
17. Products 41
18. Additive Categories 45
19. Exact Additive Categories 49
20. Abelian Categories 50
21. The Category of Abelian Groups G 51
Exercises 53
Chapter II. Diagrams and Functors 58
Introduction 58
1. Diagrams 59
2. Limits 61
3. Functors 66
4. Preservation Properties of Functors 67
5. Morphism Functors 70
6. Limit Preserving Functors 71
7. Faithful Functors 73
8. Functors of Several Variables 75
9. Natural Transformations 76
10. Equivalence of Categories 78
11. Functor Categories 80
12. Diagrams as Functors 82
13. Categories of Additive Functors Modules
14. Projectives, Injectives 86
15. Generators 88
16. Small Objects 91
Exercises 93
Chapter III. Complete Categories 98
Introduction 98
1. C1 Categories 98
2. Injective Envelopes 103
3. Existence of Injectives 105
Exercises 107
Chapter IV. Group Valued Functors 110
Introduction 110
1. Metatheorems 110
2. The Group Valued Imbedding Theorem 114
3. An Imbedding for Big Categories 118
4. Characterization of Categories of Modules 121
5. Characterization of Functor Categories 123
Exercises 129
Chapter V. Adjoint Functors 134
Introduction 134
1. Generalities 134
2. Conjugate Transformations 139
3. Existence of Adjoints 141
4. Functor Categories 143
5. Reflections 145
6. Monosubcategories 148
7. Projective Classes 153
Exercises 156
Chapter VI. Applications of Adjoint Functors 158
Introduction 158
1. Application to Limits 158
2. Module-Valued Adjoints 159
3. The Tensor Product 159
4. Functor Categories 162
5. Derived Functors 164
6. The Category of Kernel, Preserving Functors 167
7. The Full Imbedding Theorem 168
8. Complexes 169
Exercises 173
Chapter VII. Extensions 178
Introduction 178
1. Ext1 178
2. The Exact Sequence (Special Case) 184
3. Ext" 187
4. The Relation ~ 192
5. The Exact Sequence 195
6. Global Dimension 196
7. Appendix: Alternative Description of Ext 199
Exercises 204
Chapter VIII. Satellites 208
Introduction 208
1. Connected Sequences of Functors 208
2. Existence of Satellites 211
3. The Exact Sequence 216
4. Satellites of Group Valued Functors 221
5. Projective Sequences 224
6. Several Variables 225
Exercises 228
Chapter IX. Global Dimension 232
Introduction 232
1. Free Categories 232
2. Polynomial Categories 236
3. Grassmann Categories 237
4. Graded Free Categories 238
5. Graded Polynomial Categories 241
6. Graded Grassmann Categories 242
7. Finite Commutative Diagrams 244
8. Homological Tic Tac Toe 246
9. Normal Subsets 250
10. Dimension for Finite Ordered Sets 251
Exercises 258
Chapter X. Sheaves 262
Introduction 262
1. Preliminaries 262
2. F-Categories 265
3. Associated Sheaves 267
4. Direct Images of Sheaves 268
5. Inverse Images of Sheaves 270
6. Sheaves in Abelian Categories 271
7. Injective Sheaves 274
8. Induced Sheaves 275
Exercises 278
Bibliography 284
Subject Index 286