Abstract

The canonical place to interpret a first-order theory is the category of sets considering the well-behaved interaction between the syntax of the theory and the semantics that it provides. For more general theories (e.g. infinitary ones) this interaction is not so well-behaved, but by working in a more general setting we may find a suitable generalised universe of sets, i.e. a topos, where analogous interactions may be found. Thus Topos Theory can be thought of as a generalised Model Theory, where the classifying topos of a theory generalises the role that the category of sets plays for first-order theories.

I will assume the audience is familiar with the notions of adjunction and categorical limit, and will focus at least half of the talk on introducing the notion of topos along with many examples since this material shall serve as pre-requisite for many of the other talks given by the Category Theory group.