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Question: Is there any book of topology in the modern language of topos theory?

Motivation:

  • 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.

  • Relations between geometry and logic.

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    $\begingroup$ What topics are covered by "topology" to you? What do you want to see in this book? Have you read "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk? $\endgroup$
    – Steven Gubkin
    Commented Aug 30, 2014 at 17:00
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    $\begingroup$ @StevenGubkin, I think a more appropriate question is: "have you read it backward?" (there are a lot of good books about "topos from topology point of view"; so I guess, it suffices to pick any and read it backward :-) $\endgroup$
    – Michal R. Przybylek
    Commented Aug 30, 2014 at 17:46
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    $\begingroup$ If I recall correctly, Vickers, Topology Via Logic covers this kind of thing. $\endgroup$
    – aws
    Commented Aug 30, 2014 at 17:52
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    $\begingroup$ Pretty much any large enough book on topos theory is bound to describe its connection to topology. A good list of references (of vastly varying difficulty) is at the nLab page. However I believe that topoi are much more connected to logic than to topology. $\endgroup$
    – Anton Fetisov
    Commented Aug 31, 2014 at 21:06
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    $\begingroup$ I would say there is no such book in existence, but it sounds like you'd be interested in "pointless topology", i.e., locale theory. I'd recommend picking up Johnstone's Stone Spaces which, although it doesn't have locale theory as its primary aim, develops a great deal of it inter alia and has a lot of useful references. Another thing you might find interesting is Joyal and Tierney's An Extension of the Galois Theory of Grothendieck which takes seriously the analogy between locales and toposes (as lex total objects) and their respective theories of descent. $\endgroup$
    – Todd Trimble
    Commented Oct 8, 2014 at 0:42

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"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.

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    $\begingroup$ Welcome to MO! $~~~$ $\endgroup$
    – Vidit Nanda
    Commented Aug 31, 2014 at 16:54

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