First Order Logic with Isomorphism
The Univalent Foundations requires a logic that allows us to define structures on homotopy types, similar to how first order logic with equality \text{FOL}_= allows us to define structures on sets. We develop the syntax, semantics and deductive system for such a logic, which we call first order logic with isomorphism \text{FOL}_{\cong} The syntax of \text{FOL}_{\cong} extends \text{FOL}_{ in two ways. First, by incorporating into its signatures a notion of dependent sorts along the lines of Makkai's FOLDS as well as a notion of an $h$ level of each sort. Second, by specifying three new logical sorts within these signatures: isomorphism sorts, reflexivity predicates and transport structure. The semantics for \text{FOL}_{\cong} are then defined in homotopy type theory with the isomorphism sorts interpreted as identity types, reflexivity predicates as relations picking out the trivial path, and transport structure as transport along a path. We then define a deductive system \mathcal{D}_{\cong} for \text{FOL}_{\cong} that encodes the sense in which the inhabitants of isomorphism sorts really do behave like isomorphisms and prove soundness of the rules of \mathcal{D}_{\cong} with respect to its homotopy semantics. Finally, as an application, we prove that precategories, strict categories and univalent categories are axiomatizable in \text{FOL}_{\cong}