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I am currently reading Vickers' text "topology via logic" and Peter Johnstone's "stone spaces", and I understand the material in both of these texts to pertain directly to constructions in elementary topos theory (by which I do not mean 'the theory of elementary topoi). However, these things do not seem to be mentioned explicitly in these texts, at least not to great extent. Where might I avail myself of material which really 'brings home' the notion of topoi as 'generalized spaces' in the context of stone spaces and locales as alluded to in Vickers and Johnstone? I understand that Borceaux's third volume in the 'handbook of categorical algebra' is probably a good place to start...

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Many good books have already been mentioned; I like MacLane+Moerdijk as an introduction, and after that both books by Johnstone (in particular, Part C of the Elephant does a good job of connecting locale theory with topos theory). But I also wanted to mention Vickers' paper "Locales and Toposes as Spaces," which I think does a good job of connecting up the topology with the toposes and the logic in a way that isn't directly evident in many other introductions.

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I cannot praise MacLane and Moerdijk's book "Sheaves in geometry and logic" highly enough.

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    $\begingroup$ I am afraid to start reading it ... $\endgroup$
    – supercooldave
    Commented Jun 5, 2010 at 19:56
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    $\begingroup$ M&M give the most convoluted construction of the adjunction between the direct and inverse image sheaves I have ever seen in my entire life. $\endgroup$
    – Harry Gindi
    Commented Jun 5, 2010 at 23:12
  • $\begingroup$ This book is tough for those without the right kind of mathematical background. It took me a long time to work through chapter two. $\endgroup$
    – Charles Stewart
    Commented Jun 8, 2010 at 9:21
  • $\begingroup$ I agree. I would say that you should have Categories for the Working mathematician down pat first (At least up through the section on adjoint functors, and knowing about monads is very helpful as well.) $\endgroup$
    – Steven Gubkin
    Commented Jun 8, 2010 at 15:00
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The book by Robert Goldblatt: "Topoi, the Categorial Analysis of Logic" (1984) is quite simple to follow and have the nicely consulted at this address: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3

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My article "Locales and toposes as spaces", in the Handbook of Spatial Logics, serves as a readers' guide to Mac Lane and Moerdijk to show how the results there can be used to create Grothendieck's concept of toposes as generalized spaces.

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The texts by Vickers and Johnstone are rather different, and certainly are different in intention. I was struck by a remark made to me by a leading computer scientist, to the effect that Stone Spaces was a "treatise on extensionality". Mostly mathematicians, apart from some logicians, don't have much use for extensionality as a concept, and when they find out what it is, that is in the "speaking prose all my life" sense of "is that all?" But it is potentially a useful concept for understanding Grothendieck's approach in the large: not just topos theory, but also the "Yoneda lemma" approach to points and having enough of them for geometry, and the derived category approach in which spectral sequences ("intensionality") have disappeared or at least been submerged into computational undergrowth.

Naturally understanding topos theory depends on the "geometric morphism" concept. All preparatory works can do is make the consequences of this concept clearer in the more familiar context of topological spaces.

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My first "Topos Theory" book was Johnstone (some title), was hard but page after page I assimilated this (I'm still alive more or less), but it was the only book in argument.

For me opinion the Borceaux's third volume is very good

I indicate these text in progressive difficulty and depth :

S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic. A First Introduction to. Topos (a large part (localic topos) about locales and topos.

If you want Peter T. Johnstone (1977) Topos Theory Cap.7 (about a taste of spatial topos)

Peter T. Johnstone (1977) Topos Theory Sketches of an Elephant: A Topos Theory Compendium 2 (Cap. C1)

The last is the better for your request (but a bit heavy).

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