Bicategories, Biequivalence, and Bi-Interpretability

TL;DR

This paper establishes a categorical framework linking syntax and syntactic categories in coherent first-order logic, providing a precise characterization of bi-interpretability through biequivalence of bicategories.

Contribution

It introduces a categorical characterization of bi-interpretability by creating a biequivalence between bicategories of theories and categories, extending to fragments of first-order logic.

Findings

Biequivalence between bicategory of theories and coherent categories.

Necessary and sufficient conditions for bi-interpretability.

Clarification of the relation between syntax and syntactic categories.

Abstract

We make explicit the correspondence between syntax and syntactic categories for coherent first-order logic, providing a categorical characterization of bi-interpretability. This is done by creating a biequivalence between a bicategory of coherent theories and the (strict) bicategory of coherent categories. While the biequivalence concerns the stronger equality-preserving bi-interpretability, we use it to obtain a necessary and sufficient condition for two theories to be bi-interpretable in general, by relating the exact completions of their syntactic categories. These results extend analogously to familiar fragments of first-order logic, thereby clarifying the long-intuited relation between logical syntax and syntactic categories.