Topos Theory (Dover Books on Mathematics) (英語) ペーパーバック – 2014/1/15
Kindle 端末は必要ありません。無料 Kindle アプリのいずれかをダウンロードすると、スマートフォン、タブレットPCで Kindle 本をお読みいただけます。
One of the best books on a relatively new branch of mathematics, this text is the work of a leading authority in the field of topos theory. Suitable for advanced undergraduates and graduate students of mathematics, the treatment focuses on how topos theory integrates geometric and logical ideas into the foundations of mathematics and theoretical computer science.
After a brief overview, the approach begins with elementary toposes and advances to internal category theory, topologies and sheaves, geometric morphisms, and logical aspects of topos theory. Additional topics include natural number objects, theorems of Deligne and Barr, cohomology, and set theory. Each chapter concludes with a series of exercises, and an appendix and indexes supplement the text.
Mac Lane and Moerdijk SHEAVES IN GEOMETRY AND LOGIC is a beautifully written book, a long and well motivated book packed with well chosen clearly explained examples. Both those authors have a rare gift for conveying an insider's view of the subject from the start. My own book ELEMENTARY CATEGORIES, ELEMENTARY TOPOSES goes quickly and simply to the key ideas (if I say so myself). Barr and Wells TOPOSES, TRIPLES, AND THEORIES uses the most efficient tools to get at the central theorems.
But no other book goes as concisely and comprehensively to all the aspects of toposes as this one. Category theory, algebra, logic, arithmetic, geometry, and cohomology all come in, in a well chosen perspective. This book is hard to read in places, especially at the start. It cannot serve alone as an introduction unless you are really gifted. But it remains the best single text on the subject.
Johnstone has a three-volume set on the current state of topos theory due to appear later this year. It may well become the standard reference. But it will not replace this book.