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Topos theory has led to unexpected connections between classical and constructive mathematics. This text explores Lawvere and Tierney's concept of topos theory, a development in category theory that unites important but seemingly diverse notions from algebraic geometry, set theory, and intuitionistic logic. A virtually self-contained introduction, this volume presents toposes as the models of theories — known as local set theories — formulated within a typed intuitionistic logic.
The introductory chapter explores elements of category theory, including limits and colimits, functors, adjunctions, Cartesian closed categories, and Galois connections. Succeeding chapters examine the concept of topos, local set theories, fundamental properties of toposes, sheaves, locale-valued sets, and natural and real numbers in local set theories. An epilogue surveys the wider significance of topos theory, and the text concludes with helpful supplements, including an appendix, historical and bibliographical notes, references, and indexes.
About the Author
J. L. Bell is a Professor at the University of Western Ontario and co-author of Dover's Models and Ultraproducts.
Table of Contents
1. Elements of category theory
2. Introducing toposes
3. Local set theories
4. Fundamental properties of toposes
5. From logic to sheaves
6. Locale-valued sets
7. Natural numbers and real numbers
8. Epilogue: the wider significance of topos theory
Appendix: Geometric theories and classifying toposes
Historical and bibliographical notes
Index of symbols
Index of terms