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 (HoTT), on the category of Cartesian cubical sets with one connection. We thereby identify a second model structure which both constructively models HoTT and presents $\infty $ -groupoids, the first example being the equivariant Cartesian model of Awodey–Cavallo–Coquand–Riehl–Sattler.