Toposes and Local Set Theories : An Introduction

1. Elements of category theory 1 Categories 1 Some basic category-theoretic notions 5 Limits and colimits 11 Existence of limits and colimits 16 Functors 18 Natural transformations and functor categories 21 Completeness of functor categories 26 Equivalence of categories 28 Adjunctions 30 Units and counits of adjunctions 32 Freedom and cofreedom 35 Uniqueness of adjoints 38 Preservation of limits and colimits 39 Cartesian closed categories 42 Reflective subcategories 45 Galois connections 46 2. Introducing toposes 49 Subobjects and subobject classifiers 49 Power objects: the concept of topos 58 Setc as a topos 60 Geometric morphisms 65 3. Local set theories 68 Local languages and local set theories 68 Logic in a local set theory 73 Set theory in a local language 83 The category of sets determined by a local set theory 85 Interpreting a local language in a topos: the soundness theorem 91 The completeness theorem 103 The equivalence theorem 105 Translations and logical functors 113 Adjoining indeterminates 115 Introduction of function values 117 4. Fundamental properties of toposes 120 Some fundamental properties of toposes 120 The structure of Ω and Sub(A) in a topos 127 Slicing a topos 131 Coproducts in a topos 136 Syntactic properties of local set theories versus essentially categorical properties of toposes 141 Full theories 150 Beth Kripke Joyal semantics 155 5. From logic to sheaves 162 Truth sets, modalities, and universal closure operations 162 Sheaves 171 The sheafification functor 180 Modalized toposes 189 Modal operators and sheaves in Setcop 194 Sheaves over locales and topological spaces 199 6. Locale-valued sets 202 Locale-valued sets 202 The topos of sheaves on a topological space 209 Decidable, subconstant, and fuzzy sets 210 Boolean extensions as toposes 215 7. Natural numbers and real numbers 220 Natural numbers in local set theories 220 Real numbers in local set theories 226 The free topos 233 8. Epilogue: the wider significance of topos theory 235 From set theory to topos theory 235 Some analogies with the theory of relativity 239 The negation of constancy 242 Appendix: Geometric theories and classifying toposes 246 Historical and bibliographical notes 255 References 258 Index of symbols 261 Index of terms 264