I wrote a report on elementary topoi as one of the course requirements for MATH 460 (directed individual study in category theory). On Monday 2nd November 2020, I gave a talk to the VUW Logic Seminar about the contents of this report. The abstract is below.
Abstract: In a sense, logic is normally done in the category of sets, but one could ask what it means to do logic in another category. An (elementary) topos is a category which has enough structure to internally formalise logic. We will define topoi, and see the development of first-order logic in a topos. Time permitting, we may also see how some set-theoretic ideas can be generalised to this setting.