Generalising canonical extension to the categorical setting

Canonical extension has proven to be a powerful tool in algebraic study of propositional logics. In this paper we describe a generalisation 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, the fragment of first order logic in the connectives ∧, ∨, 0, 1 and ∃. 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 [22]. 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 C, its topos of types to its category of models (in Set).