nLab category of cubes

Idea

A category of cubes is a category of geometric shapes for higher structures in which the basic shapes are cubes of all dimensions. There are actually many different categories of cubes, depending on what sorts of operations are permitted between cubes; potential operations include:

• faces and degeneracies
• connections
• symmetries
• reversals
• diagonals

A presheaf on a category of cubes is a cubical set.

Definitions

The ordered cube category

The ordered cube category has faces and degeneracies, but no other operations.

Notation

We denote by $\square_{\leq 1}$ the category defined uniquely (up to isomorphism) by the following.

1) There are exactly two objects, which we shall denote by $I^{0}$ and $I^{1}$.

2) There are exactly two arrows $i_{0}, i_{1} : I^{0} \rightarrow I^{1}$.

3) There is exactly one arrow $p : I^{1} \rightarrow I^{0}$.

4) There are no non-identity arrows $I^{0} \rightarrow I^{0}$.

5) There are exactly two non-identity arrows $I^{1} \rightarrow I^{1}$, which are $i_{0} \circ p$ and $i_{1} \circ p$.

Remark

In particular, because of 4) in Notation , the diagram

$$\array{ I^{0} & \overset{i_{0}}{\to} & I^{1} \\ & \underset{id}{\searrow} & \downarrow p \\ & & I^{0} }$$

commutes in $\square_{\leq 1}$, and the diagram

$$\array{ I^{0} & \overset{i_{1}}{\to} & I^{1} \\ & \underset{id}{\searrow} & \downarrow p \\ & & I^{0} }$$

commutes in $\square_{\leq 1}$.

Remark

The category $\square_{\leq 1}$ is isomorphic to the category $\Delta_{\leq 1}$, i.e. it may also be described as

• The full subcategory of the simplex category $\Delta$ on the objects $[0]$ and $[1]$.

• (A skeleton of) the category of linearly ordered sets of cardinality 1 or 2.

• The indexing category for reflexive equalizers.

Remark

The category $\square_{\leq 1}$ can also be constructed by beginning with the free category on the directed graph defined uniquely by the fact that 1), 2), and 3) in Notation hold, and by the fact that there are no other non-identity arrows. One then takes a quotient of this free category which forces the diagrams in Remark to commute.

This quotient can be expressed as a colimit in the category of small categories, or, which ultimately amounts to the same, by means of the equivalence relation $\sim$ on the arrows of the free category generated by requiring that $p \circ i_{0} \sim id$ and $p \circ i_{1} \sim id$, and by requiring that $g_{1} \circ g_{0} \sim f_{1} \circ f_{0}$ if $g_{1} \sim f_{1}$ and $g_{0} \sim f_{0}$.

Definition

The category of cubes is the free strict monoidal category? on $\square_{\leq 1}$ whose unit object is $I^{0}$.

Notation

We denote the category of cubes by $\square$.

Terminology

We refer to $\square$ as the category of cubes.

Remark

It is not the case that $\square$ is the free strict monoidal category on $\square_{\leq 1}$. Rather, $\square$ is the free strict monoidal category with specified unit on $\square_{\leq 1}$, where the unit is specified to be $I^{0}$.

Notation

Let $n \geq 0$ be an integer. We often denote the object $\underbrace{I^{1} \otimes \cdots \otimes I^{1}}_{n}$ of $\square$ by $I^{n}$.

The symmetric cube category

The symmetric, or substructural, or semicartesian, or “BCH”, cube category has faces, degeneracies, and symmetries only.

The cartesian cube category

The cartesian, or “ABCFHL”, cube category has faces, degeneracies, symmetries, and diagonals.

The De Morgan cube category

The De Morgan, or “CCHM”, cube category has faces, degeneracies, symmetries, diagonals, connections, and reversals.

Expository material

For expository and other material, see category of cubes - exposition.

References

For the semicartesian cube category:

• Mike Shulman, Towards a Third-Generation HOTT Part 3 (slides, video)