Question: Is there any book of topology in the modern language of topos theory?
Motivation:
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In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the study of (the cohomology of) sheaves, and the sheaves "sited" on a given Grothendieck topology formed a topos – subsequently called a Grothendieck topos". my question is about a book of the study of this idea.
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Relations between geometry and logic.
Best Answer
"Topology via Logic" is only half way there. It is firmly rooted in classical mathematics and makes no connections with toposes.
Mac Lane and Moerdijk is a good suggestion. As for reading it backwards: that is pretty much the aim of my "Locales and Toposes as Spaces" (Chapter 8 in "Handbook of Spatial Logics" (ed. Aiello, Pratt-Hartman, van Bentham), Springer, 2007, pp. 429-496; ISBN 978-1-4020-5586-7). I wanted to guide the reader through the results in Mac Lane and Moerdijk in an order that brings out the "generalized space" idea of toposes.
I think it's fair to say that all those fall short of bringing out some of the ideas of algebraic topology that motivated Grothendieck in the first place. I don't know any books to recommend that cover that.
Steve Vickers.