# Connections between topos theory and topology

What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the theory I would like to cover, however, I would love some more nice examples and applications. Of course, I have some ideas of my own, but am open to suggestions. Thank you!

P.S.

I am also interested in perhaps learning about some new connections myself, which would be out of reach for such a course, so feel free to leave these as well, qualified as such.

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## 4 Answers

This is probably too advanced for a first course, but you might be interested in Ieke Moerdijk's monograph "Classifying Spaces and Classifying Topoi". Among other things, it uses topos theory to show what it is that the "classifying space" of a (non-groupoid) category classifies. IIRC this was an ingredient in the proof of the Madsen-Weiss theorem.

More obviously, one could say something about sheaf cohomology.

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Thanks Mike. I was aware of course of Ieke's book (he was my PhD adviser and all), but not that it was used to prove the Madsen-Weiss theorem, neat! – David Carchedi Mar 8 '13 at 11:01
You should probably verify that before asserting it as gospel truth; my memory is a bit hazy. (Ieke might also have more answers to your question if you asked him directly....) – Mike Shulman Mar 9 '13 at 5:01

Galois groups of atomic toposes, and relations to fundamental pro-groups of badly-behaved spaces.

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Sorry for the short answer- have a grant deadline today. – David Roberts Mar 5 '13 at 23:45
Np, good luck catching the deadline! After you do, I would love an elaboration. – David Carchedi Mar 5 '13 at 23:46
I'm going to expand my answer, just give me a few days! – David Roberts Mar 8 '13 at 3:19

Here is one that is not too grand.

Suppose you want to embed some (essentially small) category of spaces into a cartesian-closed category, so as to extend it to a convenient category. Then you can use a gros topos to do it. The site is your category of spaces, a family $\lbrace e_i : X_i \to Y\rbrace_i$ is covering when the $e_i$ are open embeddings and they cover $Y$ (I hope I got that right). The Yoneda lemma embeds the original category.

It is not hard to come up with interesting exercises that are not too demanding.

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Do you mean to say "Suppose you want to embed some (essentially small) category \textit{of} spaces ect.", or is "category spaces" possibly a definition that I am not familiar with? – Spice the Bird Mar 22 '13 at 2:58
I do, sorry for the confusion, I fixed it and thanks for pointing it out. – Andrej Bauer Mar 22 '13 at 13:48
No problem.:))) – Spice the Bird Mar 22 '13 at 16:00

If one is willing to stretch the question a little, then one can cite the theorem that states that the bicategory of Grothendieck toposes is equivalent to the bicategory of localic groupoids. This establishes a connection with general topology and Lie groupoids, presumably making Grothendieck toposes easier to understand for somebody who is already familiar with Lie groupoids and bibundles.

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Thanks Dmitry.I was planning on at least mentioning this in the class. By the way, it's not true on the nose: you need to restrict to "etale complete" localic groupoids. Otherwise, you get something equivalent to the localic version of topological stacks, which is larger than topoi. – David Carchedi Mar 22 '13 at 14:30
@David: Yes, completion is implied in my statement. More precisely, a morphism between two localic groupoids G and H is a bibundle from γG to γH, where γ is the completion functor, as explained in Moerdijk's paper, for example. – Dmitri Pavlov Mar 22 '13 at 16:26