A Syntactic Characterization of Morita Equivalence

TL;DR

This paper provides a purely syntactic characterization of Morita equivalence between theories, linking their definitional extensions to equivalent categories of models across various logical frameworks.

Contribution

It introduces a new syntactic notion of common definitional extension that characterizes Morita equivalence in a broad range of logical theories.

Findings

Syntactic characterization of Morita equivalence

Applicable to cartesian, regular, coherent, geometric, and first-order theories

Ensures equivalent categories of models in any Grothendieck topos

Abstract

We characterize Morita equivalence of theories in the sense of Johnstone in terms of a new syntactic notion of a common definitional extension developed by Barrett and Halvorson for cartesian, regular, coherent, geometric and first-order theories. This provides a purely syntactic characterization of the relation between two theories that have equivalent categories of models naturally in any Grothendieck topos.