A general method to construct cube-like categories and applications to homotopy theory

Abstract

In this paper, we introduce a method to construct new categories which look like "cubes", and discuss model structures on the presheaf categories over them. First, we introduce a notion of thin-powered structure on small categories, which provides a generalized notion of "power-sets" on categories. Next, we see that if a small category $\mathcal{R}$ admits a good thin-powered structure, we can construct a new category $\square(\mathcal{R})$ called the cubicalization of the category. We also see that $\square(\mathcal{R})$ is equipped with enough structures so that many arguments made for the classical cube category $\square$ are also available. In particular, it is a test category in the sense of Grothendieck. The resulting categories contain the cube category $\square$, the cube category with connections $\square^c$, the extended cubical category $\square_\Sigma$ introduced by Isaacson, and cube categories $\square_G$ symmetrized by more general group operads $G$. We finally discuss model structures on the presheaf categories $\square(\mathcal{R})^\wedge$ over cubicalizations. We prove that $\square(\mathcal{R})^\wedge$ admits a model structure such that the simplicial realization $\square(\mathcal{R})^\wedge\to SSet$ is a left Quillen functor. Moreover, in the case of $\square_G$ for group operads $G$, $\square^\wedge_G$ is a monoidal model category, and we have a sequence of monoidal Quillen equivalences $\square Set \to \square_G^\wedge\to SSet$. For example, if $G=B$ is the group operad consisting of braid groups, the category $\square^\wedge_B$ is a braided monoidal model category whose homotopy category is equivalent to that of $SSet$.