Categories of Theories and Interpretations

In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, bi-interpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We provide some examples to separate notions across categories. In contrast, we show that, in some cases, notions in different categories do coincide. E.g., we can, under such-and-such conditions, infer synonymity of two theories from their being equivalent in the sense of a coarser equivalence relation. We illustrate that the categories offer an appropriate framework for conceptual analysis of notions. For example, we provide a ‘coordinate free’ explication of the notion of axiom scheme. Also we give a closer analysis of the object-language/ meta-language distinction. Our basic category can be enriched with a form of 2-structure. We use this 2-structure to characterize a salient subclass of interpetations, the direct interpretations, and we use the 2-structure to characterize induction. Using this last characterization, we prove a theorem that has as a consequence that, if two extensions of Peano Arithmetic in the arithmetical language are synonymous, then they are identical. Finally, we study preservation of properties over certain morphisms.