Coherent and definable functors for triangulated categories

In this talk, I shall introduce coherent and definable functors for triangulated categories. The former are the purity preserving functors into finitely accessible categories with products, and generalise their namesakes as introduced by Krause. It will be shown that the restricted Yoneda embedding is the universal coherent functor. I will then introduce definable functors between triangulated categories, which will be shown to be those which preserve the pure structure. Their properties will be discussed and examples given. I will then give some applications to representation theory. This is based on joint work with Jordan Williamson.