The cube category $\Box$ encodes one of the main geometric shapes for higher structures. It is also called the cubical category, although that term can be ambiguous.
Its objects are the standard cellular “$n$-cubes”, for $n \in \mathbb{N}$ and its morphisms are all possible ways of mapping cubes to each other.
The cube category is the initial strict monoidal category $(M, \otimes, I)$ equipped with an object $int$ together with two maps $i_0, i_1: I \to int$ and a map $p: int \to I$ such that $p i_0 = 1_I = p i_1$.
Do we have a similar definiton of the globe category?
Todd: None that I know of; the globe category doesn’t carry a monoidal structure. But it reminds me that we should create an entry for Joyal’s category $\Theta$, used in his definition of weak $\omega$-category, as this cleverly combines globes and simplices.
Aleks: What about cubes with connection?
The cube category may also be described as the subcategory of $Set$ whose objects are powers $2^n$ of $2 = \{0, 1\}$, $n \geq 0$, and whose morphisms are generated by degeneracy maps $2^m \to 2^n$ which delete a coordinate and face maps which insert a 0 or 1 without modifying the order of coordinates. The cartesian product on $Set$ restricts to a monoidal product $\otimes$ on this subcategory, giving a strict monoidal category and indeed a pro. The basic face maps are the two inclusions $\delta^0, \delta^1: 1 \to 2$, the basic degeneracy is the map $\sigma: 2 \to 1$, and then the general face and degeneracy maps are
These satisfy the cubical identities:
… to be inserted …
The category of cubes described above has also been described as the restricted category of cubes (see the paper by Grandis and Mauri). It may be augmented in several directions, at various levels of doctrinal strength, as follows:
… to be completed? …
The cube category is used to define cubical sets.
The object $int$ may be thought of as the “generic interval” and the monoidal unit $I$ as a point; $x^{\otimes n}$ thus becomes the combinatorial $n$-cube. Indeed, the cubical set represented by $I$ is the standard cubical 0-cube, while the cubical set represented by $int$ is the standard cubical 1-cube.
An explicit description of the cube category by generators and relations is in section 2 of
Among all geometric shapes for higher structures cubes are best suited for describing Gray-like tensor products of higher structures: there is geometrically obvious way in which to combine the $n$-cube $[n]$ and the $m$-cube $[m]$ to the $(n+m)$-cube $[n] \otimes [m] := [n+m]$. This makes $\Box$ into a monoidal category. It also induces the canonical monoidal structure on cubical sets and then on strict omega-categories: the Crans-Gray tensor product.
The cube category is a test category. Hence cubical sets model homotopy types (see also model structure on cubical sets). While it is not a strict test category, it can be refined to the category of cubes with “cube connection”, which is. See connection on a cubical set for more details.