Abstract: We study the test model structure (as well as a “projective" variant) on several categories of cubical sets (cubical sets with or without connections / symmetries / reversals), study their monoidal properties, and establish Quillen equivalences among them and to simplicial sets and topological spaces.

With this groundwork in place, we then consider (on most of these cubical sites) cubical subdivision — an analog of simplicial subdivision (which is even better because it is strong monoidal!). This leads to a cubical analog of Kan’s $\mathrm{Ex}^\infty$ functor which we show to be a functorial fibrant replacement with good properties. As a corollary, we obtain cubical approximation theorems for various flavors of cubes.

Abstract: