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| Genre/Form: | Electronic books |
|---|---|
| Additional Physical Format: | Print version: Mac Lane, Saunders, 1909-2005. Sheaves in geometry and logic. New York : Springer-Verlag, ©1992 (DLC) 91033709 |
| Material Type: | Document, Internet resource |
| Document Type: | Internet Resource, Computer File |
| All Authors / Contributors: |
Saunders Mac Lane; Ieke Moerdijk |
| ISBN: | 0387977104 9780387977102 3540977104 9783540977100 |
| OCLC Number: | 828776278 |
| Description: | 1 online resource (xii, 627 pages) : illustrations |
| Contents: | Prologue -- Categorial Preliminaries -- I. Categories of Functors -- 1. The Categories at Issue -- 2. Pullbacks -- 3. Characteristic Functions of Subobjects -- 4. Typical Subobject Classifiers -- 5. Colimits -- 6. Exponentials -- 7. Propositional Calculus -- 8. Heyting Algebras -- 9. Quantifiers as Adjoints -- Exercises -- II. Sheaves of Sets -- 1. Sheaves -- 2. Sieves and Sheaves -- 3. Sheaves and Manifolds -- 4. Bundles -- 5. Sheaves and Cross-Sections -- 6. Sheaves as Étale Spaces -- 7. Sheaves with Algebraic Structure -- 8. Sheaves are Typical -- 9. Inverse Image Sheaf -- Exercises -- III. Grothendieck Topologies and Sheaves -- 1. Generalized Neighborhoods -- 2. Grothendieck Topologies -- 3. The Zariski Site -- 4. Sheaves on a Site -- 5. The Associated Sheaf Functor -- 6. First Properties of the Category of Sheaves -- 7. Subobject Classifiers for Sites -- 8. Subsheaves -- 9. Continuous Group Actions -- Exercises -- IV. First Properties of Elementary Topoi -- 1. Definition of a Topos -- 2. The Construction of Exponentials -- 3. Direct Image -- 4. Monads and Beck's Theorem -- 5. The Construction of Colimits -- 6. Factorization and Images -- 7. The Slice Category as a Topos -- 8. Lattice and Heyting Algebra Objects in a Topos -- 9. The Beck-Chevalley Condition -- 10. Injective Objects -- Exercises -- V. Basic Constructions of Topoi -- 1. Lawvere-Tierney Topologies -- 2. Sheaves -- 3. The Associated Sheaf Functor -- 4. Lawvere-Tierney Subsumes Grothendieck -- 5. Internal Versus External -- 6. Group Actions -- 7. Category Actions -- 8. The Topos of Coalgebras -- 9. The Filter-Quotient Construction -- Exercises -- VI. Topoi and Logic -- 1. The Topos of Sets -- 2. The Cohen Topos -- 3. The Preservation of Cardinal Inequalities -- 4. The Axiom of Choice -- 5. The Mitchell-Bénabou Language -- 6. Kripke-Joyal Semantics -- 7. Sheaf Semantics -- 8. Real Numbers in a Topos -- 9. Brouwer's Theorem: All Functions are Continuous -- 10. Topos-Theoretic and Set-Theoretic Foundations -- Exercises -- VII. Geometric Morphisms -- 1. Geometric Morphisms and Basic Examples -- 2. Tensor Products -- 3. Group Actions -- 4. Embeddings and Surjections -- 5. Points -- 6. Filtering Functors -- 7. Morphisms into Grothendieck Topoi -- 8. Filtering Functors into a Topos -- 9. Geometric Morphisms as Filtering Functors -- 10. Morphisms Between Sites -- Exercises -- VIII. Classifying Topoi -- 1. Classifying Spaces in Topology -- 2. Torsors -- 3. Classifying Topoi -- 4. The Object Classifier -- 5. The Classifying Topos for Rings -- 6. The Zariski Topos Classifies Local Rings -- 7. Simplicial Sets -- 8. Simplicial Sets Classify Linear Orders -- Exercises -- IX. Localic Topoi -- 1. Locales -- 2. Points and Sober Spaces -- 3. Spaces from Locales -- 4. Embeddings and Surjections of Locales -- 5. Localic Topoi -- 6. Open Geometric Morphisms -- 7. Open Maps of Locales -- 8. Open Maps and Sites -- 9. The Diaconescu Cover and Barr's Theorem -- 10. The Stone Space of a Complete Boolean Algebra -- 11. Deligne's Theorem -- Exercises -- X. Geometric Logic and Classifying Topoi -- 1. First-Order Theories -- 2. Models in Topoi -- 3. Geometric Theories -- 4. Categories of Definable Objects -- 5. Syntactic Sites -- 6. The Classifying Topos of a Geometric Theory -- 7. Universal Models -- Exercises -- Appendix: Sites for Topoi -- Epilogue -- Index of Notation. |
| Series Title: | Universitext |
| Responsibility: | Saunders Mac Lane, Ieke Moerdijk. |
Abstract:
An introduction to the theory of toposes which begins with illustrative examples and goes on to explain the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.
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