Toposes and Local Set Theories
Clarendon Press (Verlag)
9780198532743 (ISBN)
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In recent years category theory has come to play a significant role in the foundations of mathematics. The invention by Lawvere and Tierney of the concept of (elementary) "topos" may be considered one of the most striking developments in this regard. This concept unites, in a simple way, a number of seemingly diverse notions from algebraic geometry, set theory and intuitionistic logic and has led to the forging of new links between classical and constructive mathematics. This book is an introduction to what may be termed the logical approach to topos theory, that is, the presentation of toposes as the models of theories - the so-called "local set theories" - formulated within a typed intuitionistic logic.
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.
| Zusatzinfo | bibliography |
|---|---|
| Verlagsort | Oxford |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre |
| ISBN-13 | 9780198532743 / 9780198532743 |
| Zustand | Neuware |
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