For some logical theories, the data of the theory can be reconstructed from the group of automorphisms of a "special model". For example, the theory of dense linear orders can be reconstructed from the automorphism group of the rationals (considered as a dense linear order). However, for less well-behaved theories, we must consider "special" groupoids of models.
We know, due to Butz and Moerdijk, that for a theory with enough points, its classifying topos is a topos of sheaves on a topological groupoid. In this talk, we will give necessary and sufficient conditions for a groupoid of models of a theory to represent the classifying topos of a theory. Intuitively, this gives a topos-theoretic notion of when a theory can be reconstructed from a certain groupoid of models. This extends the contrast between localic Galois theory, embodied by the work of Joyal, Tierney, and Dubuc, and Caramello's topological Galois theory.