Connections between topos theory and topology

Question

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.

Answer 1

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.

Answer 2

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

Answer 3

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.

Answer 4

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.

Answer 5

I will copy and paste the description of chapter 2 and 3 of my master thesis.

Chapter 2 plays an important motivational role in the thesis and is aimed at pointing out geometric characteristics of a Grothendieck topos. For a topological space $X$ one can consider the category of its open sets $\mathcal{O}(X)$ that is a complete Heyting algebra, so that we have a functor $$ \text{Spaces} \to \text{cHa}^{\text{op}}. $$ We study properties of this functor concluding that there is a huge subcategory of spaces (sober ones) that embeds into $\text{cHa}^{\text{op}}$ via this functor. For a notational motivation we call Locales the opposite of cHa, $$\text{SobSpaces} \hookrightarrow \text{Locales} $$ So locales naturally are generalized (sober) spaces.

In Chapter 2 we present the notion of localic topos that is a Grothendieck topos on a locale and we prove that there is an equivalence of category between the category of locales and the category of localic toposes

\begin{matrix} \text{Locales} & \leftrightarrows & \text{LocToposes.} & \end{matrix}

This equivalence is the precise sense in which a localic topos is a generalized topological space, that is the same in which its associated locale is a generalized topological space.

A generic Grothendieck topos has not this fascinating property, there is not a topological space from which it comes from, but precisely in this rift one can collocate Barr's theorem.

Chapter 3 is devoted to proving Barr's theorem that we can formulate right now:

Any Grothendieck topos is covered by a localic boolean one.

This theorem states that not any Grothendieck topos is geometric but not far from it there is an other Grothendieck topos that is not only geometric (better say localic) but it is also boolean. This is the first and most naive interpretation of Barr's theorem.