Title: Theories classified by topological groupoids

Abstract: A classical result of model theory asserts that the topological automorphism group of a countable model characterises an omega-categorical theory up to the level of bi-interpretability – a notion of when two theories are ‘equivalent’.  Do we have to use a countable model?  Can we remove the assumption that the theory is omega-categorical?

We present an approach to these questions based on topos theory.  We will define when a theory is represented by a groupoid of its models, and demonstrate that two theories are bi-interpretable if and only if their representing groupoids are weakly equivalent.