Sheaves in Geometry and Logic

• 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 Morphismsand 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-OrderTheories.- 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.

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