Morgan Rogers: Endomorphisms of models: yet another categorification of model theory

Abstract

Model theory has inspired a wide spectrum of developments in category theory, often with a goal of recasting this very set-theoretic domain in a categorical framework. This vein still has plenty of ideas to mine, of which I shall present one originally suggested to me by the late Pieter Hofstra: the correspondence between closed subgroups of the (topological) automorphism group of a set and extensions of that set by relations. We recast this in terms of classifying toposes, and show how this enables us to restate the result with monoids in place of groups and almost arbitrary structures in place of sets.