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.
Best Answer
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.