Morphism axioms

We introduce a new concept in the area of formal logic: axioms for model morphisms. We work in the setting of specification languages that define the semantics of a theory as a category of models. While it is routine to use axioms to specify the class of models of a theory, there has so far been no analogue to systematically specify the morphisms between these models. This leads to subtle problems where it is difficult to give a theory that specifies the intended model category, or where seemingly isomorphic theories actually have non-isomorphic model categories. Our morphism axioms remedy this by providing new syntax for axiomatizing and reasoning about the properties of model morphisms. Additionally, our system resolves a subtle incompatibility between theory morphisms and model morphisms: the semantics that maps theories to model categories is functorial. While this result is standard in principle, previous formulations had to restrict the allowed theory morphisms or the allowed model morphisms. Our system allows establishing the result in full generality.