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 $\text{FOL}_{\cong}$.