Topos theory can be a little intimidating: it's a vast subject with a lot of baggage attached, and like many areas of category theory you need to be prepared to go quite deep to be rewarded. Unfortunately, that entry barrier makes the subject almost inaccessible to a casual reader! In this talk, I will help you get your head around the jargon. What is a subobject classifier? What is the difference between an elementary topos and a Grothendieck topos? Where do sheaves come in and what are they, anyway? How are these generalized spaces? What is a classifying topos? What's the internal language? Nominal sets form some kind of topos, right? What about infinity-toposes? Armed with this knowledge and a bunch of examples, you'll stand a better chance of figuring out how to make toposes work for you.

For the uninitiated, I will spend the first 15 minutes or so introducing category theory through some examples (which may miraculously turn out to be toposes). I don't expect to have a lot of time for non-examples, but I am happy to field questions. I also don't know in advance whether I will be able to cover everything I wrote above, but I will try my best.

(note for people following at a distance: the talk will be given on a blackboard)