category of cubes - exposition




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 “nn-cubes”, for nn \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,,I)(M, \otimes, I) equipped with an object intint together with two maps i 0,i 1:Iinti_0, i_1: I \to int and a map p:intIp: int \to I such that pi 0=1 I=pi 1p 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 Θ\Theta, used in his definition of weak ω\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 SetSet whose objects are powers 2 n2^n of 2={0,1}2 = \{0, 1\}, n0n \geq 0, and whose morphisms are generated by degeneracy maps 2 m2 n2^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 SetSet 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 δ 0,δ 1:12\delta^0, \delta^1: 1 \to 2, the basic degeneracy is the map σ:21\sigma: 2 \to 1, and then the general face and degeneracy maps are

δ i ε=2 i1δ ε2 ni:2 n12 n,σ i=2 i1σ2 ni:2 n2 n1\delta_i^\varepsilon = 2^{i-1} \otimes \delta^\varepsilon \otimes 2^{n-i}: 2^{n-1} \to 2^n, \qquad \sigma_i = 2^{i-1} \otimes \sigma \otimes 2^{n-i}: 2^n \to 2^{n-1}

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 intint may be thought of as the “generic interval” and the monoidal unit II as a point; x nx^{\otimes n} thus becomes the combinatorial nn-cube. Indeed, the cubical set represented by II is the standard cubical 0-cube, while the cubical set represented by intint is the standard cubical 1-cube.

  • An explicit description of the cube category by generators and relations is in section 2 of

    • Sjoerd Crans, Pasting schemes for the monoidal biclosed structure on ωCat\omega-Cat (web, ps, pdf)
  • 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 nn-cube [n][n] and the mm-cube [m][m] to the (n+m)(n+m)-cube [n][m]:=[n+m][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.


As a test category

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.

category: category

Last revised on February 3, 2019 at 10:10:36. See the history of this page for a list of all contributions to it.