Univalent universes for elegant models of homotopy types

Abstract

We construct a univalent universe in the sense of Voevodsky in some suitable model categories for homotopy types (obtained from Grothendieck's theory of test categories). In practice, this means for instance that, appart from the homotopy theory of simplicial sets, intensional type theory with the univalent axiom can be interpreted in the homotopy theory of cubical sets (with connections or not), or of Joyal's cellular sets.