Generalising canonical extension to the categorical setting

TL;DR

This paper extends the concept of canonical extension from distributive lattices to coherent categories, providing a categorical semantics for first-order logic and new insights into the topos of types.

Contribution

It introduces a universal canonical extension for coherent categories, generalizing existing lattice-based notions and linking to the topos of types and model categories.

Findings

Generalizes canonical extension to coherent categories

Provides new proofs for properties of the topos of types

Establishes a relation between the topos of types and models in Sets

Abstract

Canonical extension has proven to be a powerful tool in algebraic study of propositional logics. In this paper we describe a generalization of the theory of canonical extension to the setting of first order logic. We define a notion of canonical extension for coherent categories. These are the categorical analogues of distributive lattices and they provide categorical semantics for coherent logic. We describe a universal property of our construction and show that it generalises the existing notion of canonical extension for distributive lattices. Our new construction for coherent categories has led us to an alternative description of the topos of types, introduced by Makkai in the late seventies. This allows us to give new and transparent proofs of some properties of the action of the topos of types construction on morphisms. Furthermore, we prove a new result relating, for a coherent category, its topos of types to its category of models (in Sets).