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

Research paper by **Jun Yoshida**

Indexed on: **26 Feb '15**Published on: **26 Feb '15**Published in: **Mathematics - Category Theory**

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#### 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 $SSet$.