Definable Categorical Equivalence

This paper aims to provide a new criterion of formal equivalence of theories that is suitable for being supplemented with preservation conditions concerning interpretation. I argue that both the internal structure of models and choices of morphisms between models are aspects of formalisms that are relevant when it comes to their interpretations. So an adequate criterion should take these aspects into account. The two most important criteria presently discussed in philosophy of science---generalised definitional equivalence (Morita equivalence) and categorical equivalence---are not optimal in this respect. Generalised definitional equivalence neglects choices of morphisms, whereas categorical equivalence neglects the internal structure of models. I put forward the new criterion of definable categorical equivalence which takes into account both aspects and connects the category-theoretic approach and the definability-theoretic approach to formal equivalence