Elementary categories, elementary toposes

Contents......Page 6
1 Individual categories......Page 11
2 The category of categories......Page 15
3 Toposes......Page 16
4 Advice on reading......Page 18
Part I Categories......Page 20
1.1 Axioms......Page 21
1.2 Isomorphisms, monies, and epics......Page 22
1.4 Generalized elements......Page 24
1.5 Monies, isos, and generalized elements......Page 25
Exercises......Page 26
2.1 Commutative diagrams......Page 28
2.2 Products......Page 29
2.3 Some natural isomorphisms......Page 31
2.4 Finite products......Page 35
2.5 Co-products......Page 36
2.6 Equalizers and coequalizers......Page 38
Exercises......Page 39
3.1 Definition......Page 42
3.2 Homomorphisms......Page 43
3.3 Algebraic structures......Page 44
Exercises......Page 45
4.1 Sub-objects......Page 46
4.2 Pullbacks......Page 49
4.3 Guises of pullbacks......Page 51
4.4 Theorems on pullbacks......Page 53
4.5 Cones and limits......Page 56
4.6 Limits as equalizers of products......Page 58
Exercises......Page 59
5.2 Equivalence relations......Page 62
Exercises......Page 63
6.1 Exponentials......Page 65
6.2 Internalizing composition......Page 66
6.4 Initial objects and pushouts......Page 68
6.5 Intuitive discussion......Page 70
6.6 Indexed families of arrows......Page 71
Exercises......Page 73
7.1 Extending the language......Page 76
Exercises......Page 77
Part II The Category of Categories......Page 78
8.1 Functors......Page 79
8.3 Constructing categories from categories......Page 80
8.4 Aspects of finite categories......Page 84
Exercises......Page 85
9.1 Definition......Page 88
9.2 Functor categories......Page 89
9.3 Equivalence......Page 90
Exercises......Page 91
10.1 Universal arrows......Page 94
10.2 Adjunctions......Page 95
10.3 Proofs......Page 96
10.4 Adjunctions as isomorphisms......Page 98
10.5 Adjunctions compose......Page 100
Exercises......Page 101
11.1 Indexed families of objects......Page 105
11.2 Internal products......Page 107
11.3 Functors between slices......Page 110
Exercises......Page 111
12.1 Set-theoretic foundations......Page 113
12.2 Axiomatizing the category of categories......Page 116
Exercises......Page 118
Part III Toposes......Page 120
13.2 The sub-object classifier......Page 121
13.3 Conjunction and intersection......Page 122
13.4 Order and implicates......Page 123
13.5 Power objects......Page 124
13.6 Universal quantification......Page 125
13.7 Members of implicates and of universal quantifications......Page 126
Exercises......Page 127
14.1 The language......Page 130
14.2 Topos logic......Page 132
14.3 Proofs in topos logic......Page 134
Exercises......Page 136
15.1 Defining fa, ~, \vee, and (\exists x)......Page 139
15.2 Soundness......Page 140
Exercises......Page 143
16.1 Overview......Page 145
16.2 Monies and epics......Page 146
16.3 Functional relations......Page 147
16.5 Initial objects and negation......Page 149
16.6 Coproducts......Page 151
16.7 Equivalence relations......Page 153
16.8 Coequalizers......Page 154
Exercises......Page 155
17.1 Partial arrow classifiers......Page 158
17.2 Local Cartesian closedness......Page 160
17.3 The fundamental theorem......Page 162
17.4 Stability......Page 163
17.5 Complements and Boolean toposes......Page 165
17.6 The axiom of choice......Page 166
Exercises......Page 168
18.1 Satisfaction......Page 171
18.2 Generic elements......Page 173
Exercises......Page 175
19.1 Definition......Page 176
19.2 Peano's axioms......Page 177
19.3 Arithmetic......Page 178
19.4 Order......Page 179
19.5 Rational and real numbers......Page 181
19.6 Finite cardinals......Page 182
Exercises......Page 183
20.1 Small categories......Page 186
20.2 E-valued functors......Page 187
20.3 The Yoneda lemma......Page 191
20.4 E^A is a topos......Page 193
Exercises......Page 195
21.1 Definition......Page 200
21.2 Sheaves......Page 202
21.3 The sheaf reflection......Page 203
21.4 Grothendieck toposes......Page 209
Exercises......Page 210
Part IV Some Toposes......Page 212
22.1 Axioms......Page 213
22.2 Diagram categories over Set......Page 215
22.3 Membership-based set theory......Page 217
Exercises......Page 218
23.1 A ring of line type......Page 221
23.2 Calculus......Page 223
23.3 Models over Set......Page 226
Exercises......Page 228
24.1 Constructing the topos......Page 231
24.2 Realizability......Page 234
24.3 Features of Eff......Page 235
Exercises......Page 239
25.1 Categories of relations......Page 243
25.2 Map(C)......Page 247
25.3 When Map(C) is a topos......Page 250
Exercises......Page 253
Further reading......Page 255
Bibliography......Page 259
Symbol index......Page 264
Subject index......Page 265