Relative elegance and cartesian cubes with one connection

Abstract

We establish a Quillen equivalence between the Kan-Quillen model structure and a model structure, derived from a cubical model of homotopy type theory, on the category of cartesian cubical sets with one connection. We thereby identify a second model structure which both constructively models homotopy type theory and presents infinity-groupoids, the first example being the equivariant cartesian model of Awodey-Cavallo-Coquand-Riehl-Sattler.