Toposes and local set theories: an introduction

Description

Content: Part 1 Elements of category theory: including categories, functors, adjunctions, uniqueness of adjoints, Cartesian closed categories, reflective subcategories, Galois connections. Part 2 Introducing toposes: including geometric morphisms, power objects - the concept of topos. Part 3 Local set theories: including local languages and local set theories, interpreting a local language in a topos - the Soundness Theorem, the Completeness Theorem, the Equivalence Theorem, adjoining indeterminates, introduction of function values. Part 4 Fundamental properties of toposes: including slicing a topos, Beth-Kripke-Joyal semantics. Part 5 From logic to sheaves: including truth sets, modalities and universal closure operations, the Sheafification functor, modalized toposes, sheaves over locales and topological spaces. Part 6 Locale-valued sets: including the topos of sheaves over a topological space, decidable, subconstant and fuzzy sets, Boolean extensions as toposes. Part 7 Natural numbers and real numbers: including natural and real numbers in local set theories, the free topos. Part 8 Epilogue - the wider significance of topos theory: from set theory to topos theory, some analogies with the Theory of Relativity, the negation of constancy. Appendix - geometric theories and classifying toposes. Historical and bibliographical notes. References. Index of symbols. Index of terms.