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- W2757705450 abstract "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 $text{FOL}_{cong}$." @default.
- W2757705450 created "2017-10-06" @default.
- W2757705450 creator A5027417057 @default.
- W2757705450 date "2016-03-09" @default.
- W2757705450 modified "2023-09-27" @default.
- W2757705450 title "First-Order Logic with Isomorphism" @default.
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