Toposes and Local Set Theories: An Introduction

by J.L. Bell

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.

Genres: Mathematics

288 pages, Hardcover

First published January 1, 1988

About the author

John Bell (b. March 25, 1945) is professor of Logic and the Philosophy of Mathematics at the University of Western Ontario in Canada. In 2006-07, he was named the first Graham and Gail Wright Faculty of Arts Distinguished Scholar at the University of Western Ontario. In 2009, he was elected a Fellow of the Royal Society of Canada. He was admitted on a scholarship to Oxford University at the age of 15, and graduated with a D.Phil. in Mathematics at the age of 21. His dissertation supervisor was John Crossley.[1]

He was appointed assistant lecturer in the Mathematics Department at the London School of Economics in 1968, and was appointed reader in Mathematical Logic in 1980. He taught at LSE until 1989. During this time, he served as visiting fellow at the Polish Academy of Sciences (1975) and National University of Singapore (1980, 1982). In 1989, he took a position as professor in the Philosophy Department at UWO. He is also an adjunct professor in the Mathematics Department at UWO.[1]

John Bell's students include Graham Priest (Ph.D. Mathematics LSE, 1972), Michael Hallet (Ph.D. Philosophy LSE, 1979), Elaine Landry (Ph.D. Philosophy UWO, 1997) and David DeVidi (Ph.D. Philosophy UWO, 1994).

http://en.wikipedia.org/wiki/John_Lan...

Reviews

[Tim Robinson · Rating 3 out of 5]

Too difficult. If you have not already mastered Category Theory, this is not the place to start.


Too many proofs and not enough examples. To introduce a topos as a special kind of category conceals its true nature. You wouldn't introduce set theory in this way! And in fact, topos theory is merely a generalisation of set theory that allows extra truth values besides TRUE and FALSE. It might be better to discuss toposes without any reference to category theory at all.