Henrik Forssell: Recovering theories from their models

Given a theory T and its category of models and homomorphisms Mod(T), is it possible to recover T from Mod(T) (up to some suitable notion of equivalence)? A positive answer for regular theories was given by Makkai who showed that the classifying topos of a regular theory - from which the theory can be recovered - can be represented as filtered colimit preserving functors from Mod(T) to the category SET of sets and functions. Moving to coherent (and classical first-order theories), however, it becomes necessary to equip Mod(T) with some extra structure. While Makkai uses structure based on ultra-products for this case, it is possible to equip Mod(T) with a natural topology and represent the classifying topos of T as equivariant sheaves on the resulting topological category (or groupoid, considering just the isomorphisms). This forms the basis of an extension of Stone duality to first-order theories, and allows for the application of topos-theoretic techniques to e.g. give a topological characterization of the definable subclasses of Mod(T).