The Univalence Axiom in posetal model categories

Abstract

Abstract. In this note we interpret Voevodsky's Univalence Axiom in the language of (abstract) model categories. We then show that any posetal locally Cartesian closed model category Qt in which the mapping Hom (w) (Z × B, C) : Qt −→ Sets is functorial in Z and represented in Qt satisfies our homotopy version of the Univalence Axiom, albeit in a rather trivial way. This work was motivated by a question reported in [Gar11], asking for a model of the Univalence Axiom not equivalent to the standard one.