Topoi (eBook)

Front Cover 1
Topoi: The Categorial Analysis of Logic 4
Copyright Page 5
Table of Contents 16
Dedication 6
PREFACE 10
PREFACE TO SECOND EDITION 15
PROSPECTUS 18
CHAPTER 1. MATHEMATICS = SET THEORY? 23
1. Set theory 23
2. Foundations of mathematics 30
3. Mathematics as set theory 31
CHAPTER 2. WHAT CATEGORIES ARE 34
1. Functions are sets? 34
2. Composition of functions 37
3. Categories: first examples 40
4. The pathology of abstraction 42
5. Basic examples 43
CHAPTER 3. ARROWS INSTEAD OF EPSILON 54
1. Monic arrows 54
2. Epic arrows 56
3. Iso arrows 56
4. Isomorphic objects 58
5. Initial objects 60
6. Terminal objects 61
7. Duality 62
8. Products 63
9. Co-products 71
10. Equalisers 73
11. Limits and co-limits 75
12. Co-equalisers 77
13. The 
80 
14. Pushouts 85
15. Completeness 86
16. Exponentiation 87
CHAPTER 4. INTRODUCING TOPOI 92
1. Subobjects 92
2. Classifying subobjects 96
3. Definition of topos 101
4. First examples 102
5. Bundles and sheaves 105
6. Monoid actions 117
7. Power objects 120
8. O 
124 
CHAPTER 5. TOPOS STRUCTURE: FIRST STEPS 126
1. Monies equalise 126
2. Images of arrows 127
3. Fundamental facts 131
4. Extensionality and bivalence 132
5. Monies and epics by elements 140
CHAPTER 6. LOGIC CLASSICALLY CONCEIVED 142
1. Motivating topos logic 142
2. Propositions and truth-values 143
3. The prepositional calculus 146
4. Boolean algebra 150
5. Algebraic semantics 152
6. Truth-functions as arrows 153
7. 
157 
CHAPTER 7. ALGEBRA OF SUBOBJECTS 163
1. Complement, intersection, union 163
2. Sub(d) as a lattice 168
3. Boolean topoi 173
4. Internal vs. external 176
5. Implication and its implications 179
6. Filling two gaps 183
7. Extensionality revisited 185
CHAPTER 8. INTUITIONISM AND ITS LOGIC 190
1. Constructivist philosophy 190
2. Heyting's calculus 194
3. Heyting algebras 195
4. Kripke semantics 204
CHAPTER 9. FUNCTORS 211
1. The concept of functor 211
2. Natural transformations 215
3. Functor categories 219
CHAPTER 10. SET CONCEPTS AND VALIDITY 228
1. Set concepts 228
2. Heyting algebras in P 230
3. The subobject classifier in 
232 
4. The truth arrows 238
5. Validity 240
6. Applications 244
CHAPTER 11. ELEMENTARY TRUTH 247
1. The idea of a first-order 
247 
2. Formal language and 
251 
3. Axiomatics 254
4. Models in a topos 255
5. Substitution and soundness 266
6. Kripke models 273
7. Completeness 281
8. Existence and free logic 283
9. Heyting-valued sets 291
10. High-order logic 303
CHAPTER 12. CATEGORIAL SET THEORY 306
1. Axioms of choice 307
2. Natural numbers objects 318
3. Formal set theory 322
4. Transitive sets 330
5. Set-objects 337
6. Equivalence of models 345
CHAPTER 13. ARITHMETIC 349
1. Topoi as foundations 349
2. Primitive recursion 352
3. Peano postulates 364
CHAPTER 14. LOCAL TRUTH 376
1. Stacks and sheaves 376
2. Classifying stacks and sheaves 385
3. Grothendieck 
391 
4. Elementary sites 395
5. Geometric modality 398
6. Kripke–Joyal 
403 
7. Sheaves as complete 
405 
8. Number systems as sheaves 430
CHAPTER 15. ADJOINTNESS AND QUANTIFIERS 455
1. Adjunctions 455
2. Some adjoint situations 459
3. The fundamental theorem 466
4. Quantifiers 470
CHAPTER 16. LOGICAL GEOMETRY 475
1. Preservation and reflection 476
2. Geometric morphisms 480
3. Internal logic 500
4. Geometric logic 510
5. Theories as sites 521
REFERENCES 538
CATALOGUE OF NOTATION 548
INDEX OF DEFINITIONS 558