Symmetric cubical sets

Abstract

We introduce a new cubical model for homotopy types. More precisely, we will define a category Q Σ with the following features: Q Σ is a prop containing the classical box category as a subcategory; the category q Σ Set of presheaves of sets on Q Σ models the homotopy category; and combinatorial symmetric monoidal model categories with cofibrant unit have homotopically well-behaved q Σ Set enrichments.