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In this paper, we introduce a method to construct new categories which looklike "cubes", and discuss model structures on the presheaf categories overthem. First, we introduce a notion of thin-powered structure on smallcategories, which provides a generalized notion of "power-sets" on categories.Next, we see that if a small category $\mathcal{R}$ admits a good thin-poweredstructure, we can construct a new category $\square(\mathcal{R})$ called thecubicalization of the category. We also see that $\square(\mathcal{R})$ isequipped with enough structures so that many arguments made for the classicalcube category $\square$ are also available. In particular, it is a testcategory in the sense of Grothendieck. The resulting categories contain thecube category $\square$, the cube category with connections $\square^c$, theextended cubical category $\square_\Sigma$ introduced by Isaacson, and cubecategories $\square_G$ symmetrized by more general group operads $G$. Wefinally 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 simplicialrealization $\square(\mathcal{R})^\wedge\to SSet$ 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 Quillenequivalences $\square Set \to \square_G^\wedge\to SSet$. 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 tothat of $SSet$.

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