Errol Yuksel: Toposes vs Localic Groupoids

Abstract

I will report on work in progress (joint with Ivan Di Liberti & Peter LeFanu Lumsdaine) aiming to unify the different representation theorems of toposes found in the literature.

A representation theorem is one that takes each topos E to a groupoid in the category of locales presenting E. I will introduce the notion of amorphous sheaves, these are particularly concrete objects: a locale and a sheaf over it satisfying some conditions. This definition relies on the notion of descent morphism for toposes, which will be briefly recalled.

I will then show that each amorphous sheaf generates a representation theorem. This is a relatively involved categorical construction so only the conceptually important steps will be sketched. For instance, some time will be spent to introduce an intermediate notion, that of a covering functor.

Time allowing, I will end by presenting a recognition principle for amorphous sheaves, which is based on the logical nature of toposes.