Symmetric cubical sets

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