The classical Stone duality between boolean algebras and Stone spaces and the Gabriel-Ulmer duality between small finitely complete categories and locally finitely presentable categories are in fact the syntax-semantics dualities for propositional logic and Cartesian fragment of the first order logic respectively. For regular logic, such a duality between definable additive categories and small exact categories was proven by Prest and Rajani. The proof of this result heavily relies on the tools specific to additive as well as abelian categories and the question was motivated in the model theory of modules.
In a recent work with Rosicky we extended this duality to the non-additive setting using new methods and demonstrated that the techniques could also be used to give a new proof of the Prest-Rajani duality. In the talk I will give an overview of the history of the problem and explain the statement as well as the intricacies of the non-additive case.