On topological groupoids that represent theories

Grothendieck toposes, and by extension, logical theories, can be represented by topological structures. Butz and Moerdijk showed that every topos with enough points can be represented as the topos of sheaves on an open topological groupoid. This paper tackles a follow-up question: we characterise, in model-theoretic terms, which open topological groupoids can represent the classifying topos of a theory. Intuitively, this characterises which groupoids of models contain enough information to reconstruct the theory. Our treatment subsumes many of the previous approaches found in the literature, such as that of Awodey, Forssell, Butz and Moerdijk.