2016/03/09 by Dimitris Tsementzis, Tsementzis, Dimitris
Computer Science · Mathematics · #03B15 #03B22 #03C99 #03G99 #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Logic, Reasoning, and Knowledge #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.1603.03092
openalex publication_date 2016/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Univalent Foundations requires a logic that allows us to define structures on homotopy types, similar to how first-order logic with equality (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 (FOL≅). The syntax of FOL≅ extends 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 FOL≅ 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 D≅ for FOL≅ that encodes the sense in which the inhabitants of isomorphism sorts really do behave like isomorphisms and prove soundness of the rules of D≅ with respect to its homotopy semantics. Finally, as an application, we prove that precategories, strict categories and univalent categories are axiomatizable in FOL≅.