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Sheaves in geometry and logic : a first introduction to topos theory

Author: Saunders Mac Lane; Ieke Moerdijk
Publisher: New York : Springer-Verlag, ©1992.
Series: Universitext
Edition/Format:  eBook : Document : EnglishView all editions and formats
Summary:
We dedicate this book to the memory of J. Frank Adams. His clear insights have inspired many mathematicians, including both of us. In January 1989, when the first draft of our book had been completed, we heard the sad news of his untimely death. This has cast a shadow on our subsequent work. Our views of topos theory, as presented here, have been shaped by continued study, by conferences, and by many personal  Read more...
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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  Read more...

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Frank Adams. His clear insights have inspired many mathematicians, including both of us. In January 1989, when the first draft of our book had been completed, we heard the sad news of his untimely death. This has cast a shadow on our subsequent work. Our views of topos theory, as presented here, have been shaped by continued study, by conferences, and by many personal contacts with friends and colleagues-including especially O. Bruno, P. Freyd, J.M.E. Hyland, P.T. Johnstone, A. Joyal, A. Kock, F.W. Lawvere, G.E. Reyes, R Solovay, R Swan, RW. Thomason, M. Tierney, and G.C. Wraith. Our presentation combines ideas and results from these people and from many others, but we have not endeavored to specify the various original sources. Moreover, a number of people have assisted in our work by pro\u00AD viding helpful comments on portions of the manuscript. In this respect, we extend our hearty thanks in particular to P. Corazza, K. Edwards, J. Greenlees, G. Janelidze, G. Lewis, and S. Schanuel.<\/span>\"@en<\/a> ;\u00A0\u00A0\u00A0\nschema:description<\/a> \"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 \u00C9tale 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-B\u00E9nabou 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. 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