Famously, Joyal and Tierney proved that every Grothendieck topos can be represented as a topos of equivariant sheaves over a localic groupoid. Hence, toposes can be understood as ‘locales plus automorphisms’. However, it appears that non-experts are often unsure about exactly how to construct such a localic groupoid in concrete examples. In this talk I will explain how to write down the resulting localic groupoid directly from a geometric theory classified by the topos in question. In fact, we find that the groupoid comes equipped with a generic étale bundle which classifies models of the theory, not just in (localic groupoids arising from) toposes, but over more general localic groupoids. This perspective suggests the possibility of analogous constructions that go beyond the setting of toposes, such a localic groupoid that classifies proper separated bundles.

The talk is based on joint work with Joshua Wrigley.