Contents

# Contents

## Definition

###### 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 $$ and $$.

• (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

###### 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 $I^{n}$.

## Variants

There are several useful variations of $\square$, to be described on other pages in the future.

## Expository material

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

Last revised on February 3, 2019 at 10:06:33. See the history of this page for a list of all contributions to it.