Sheaves in Geometry and Logic

This book is an introduction to the theory of toposes, as first developed by Grothendieck and later developed by Lawvere and Tierney. Beginning with several illustrative examples, the book explains 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. This is the first text to address all of these various aspects of topos theory at the graduate student level.

This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds (algebraic, analytic, etc.). Sheaves also appear in logic as carriers for models of set theory as well as for the semantics of other types of logic. Grothendieck introduced a topos as a category of sheaves for algebraic geometry. Subsequently, Lawvere and Tierney obtained elementary axioms for such (more general) categories.

Autorentext

Biography of Saunders Mac Lane Saunders Mac Lane was born on August 4, 1909 in Connecticut. He studied at Yale University and then at the University of Chicago and at Göttingen, where he received the D.Phil. in 1934. He has tought at Harvard, Cornell and the University of Chicago. Mac Lane's initial research was in logic and in algebraic number theory (valuation theory). With Samuel Eilenberg he published fifteen papers on algebraic topology. A number of them involved the initial steps in the cohomology of groups and in other aspects of homological algebra - as well as the discovery of category theory. His famous and undergraduate textbook Survey of modern algebra, written jointly with G. Birkhoff, has remained in print for over 50 years. Mac Lane is also the author of several other highly successful books.

Zusammenfassung
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. Our presentation combines ideas and results from these people and from many others, but we have not endeavored to specify the various original sources.

Inhalt
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.