Cubical homotopy theory and monoidal model categories

Suppose C is a combinatorial symmetric monoidal model category. Dan Dugger has shown that C may be realized as a left Bousfield localization of the projective model structure on simplicial presheaves on a small site. However, if C is not already simplicially enriched, this presentation will not respect the monoidal structure of C . In this paper we will construct a symmetric cubical site QS extending the classical cubical site. By replacing the category of simplicial sets with the category of presheaves of sets over QS we can use Dugger's methods to produce a presentation of C as presheaves of spaces on a monoidal site retaining the monoidal structure in C as the convolution product.