Simplicial sets inside cubical sets

Abstract

As observed by various people recently the topos sSet of simplicial sets appears as essential subtopos of a topos cSet of cubical sets, namely presheaves over the category FL of finite lattices and monotone maps between them. The latter is a variant of the cubical model of type theory due to Cohen et al. for the purpose of providing a model for a variant of type theory which validates Voevodsky's Univalence Axiom and has computational meaning.One easily shows that sheaves, i.e. simplicial sets, are closed under most of the type theoretic operations of cSet. Finally, we construct in cSet a fibrant univalent universe for those types that are sheaves.