Variations on Analytic Functors

This talk will be a survey of results around the notion of an analytic functor. We will begin by recalling the notion of an analytic functor on the category of sets, which are functorial counterparts of exponential power series, originally defined by Joyal as part of his work on species of structures. We will then introduce and discuss two successive generalisations of this notion. First, analytic functors between presheaf categories (introduced in joint work with Fiore, Hyland and Winskel); secondly, analytic functors between categories of modules for a coloured operad (introduced in joint work with Joyal). The main results of the talk will be that both of these generalised notions give rise to cartesian closed 2-categories.