nLab homotopy groups of a cubical Kan complex

A relation on the $n$-cubes of a cubical set with trivial boundary

We make use of the notation established at cubical set and category of cubes.

Throughout this page, we shall let $X$ be a cubical set, let $x : \square^{0} \rightarrow X$ be a $0$-cube of $X$, and let $n \geq 0$ be an integer.

Notation

We denote by $Z_{n}(X,x)$ the set of $n$-cubes $\sigma : \square^{n} \rightarrow X$ of $X$ with the property that the following diagram in $\mathsf{Set}^{\square^{op}}$ commutes for every integer $1 \leq i \leq n$ and every integer $0 \leq \epsilon \leq 1$.

$$\array{ \square^{n-1} & \overset{p}{\rightarrow} & \square^{0} \\ \mathllap{y(I^{i-1} \otimes i_{\epsilon} \otimes I^{n-i})} \downarrow & & \downarrow \mathrlap{x} \\ \square^{n} & \underset{\sigma}{\rightarrow} & X }$$

Notation

Let $\sim$ be the relation on $Z_{n}(X,x)$ given by identifying $\sigma_{0}$ and $\sigma_{1}$ if there is an $(n+1)$-cube $h : \square^{n+1} \rightarrow X$ of $X$ such that the following diagrams in $\mathsf{Set}^{\square^{op}}$ commute

$$\array{ \square^{n} & \overset{\square^{n} \otimes i_{0}}{\rightarrow} & \square^{n+1} \\ & \underset{\sigma_{0}}{\searrow} & \downarrow h \\ & & X }$$

$$\array{ \square^{n} & \overset{\square^{n} \otimes i_{1}}{\rightarrow} & \square^{n+1} \\ & \underset{\sigma_{1}}{\searrow} & \downarrow h \\ & & X }$$

and such that the following diagram in $\mathsf{Set}^{\square^{op}}$ commutes for every integer $1 \leq i \leq n$ and every integer $0 \leq \epsilon \leq 1$.

$$\array{ \square^{n} & \overset{p}{\rightarrow} & \square^{0} \\ \mathllap{y(I^{i-1} \otimes i_{\epsilon} \otimes I^{n+1-i})} \downarrow & & \downarrow \mathrlap{x} \\ \square^{n+1} & \underset{\sigma}{\rightarrow} & X }$$

Remark

The commutativity of the first two diagrams in Notation asserts that $h$, viewed as an arrow $\square^{n} \otimes \square^{1} \rightarrow X$ of $\mathsf{Set}^{\square^{op}}$, defines a homotopy? from $\sigma_{0} : \square^{n} \rightarrow X$ to $\sigma_{1} : \square^{n} \rightarrow X$.

The relation $\sim$ is in fact an equivalence relation

Proposition

Let $\sigma$ be an $n$-cube of $X$ which belongs to $Z_{n}(X,x)$. Then $\sigma \sim \sigma$.

Proof

We take $h$ to be the arrow $\sigma \circ \big( \square^{n} \otimes y(p) \big) : \square^{n+1} \rightarrow \square^{n} \rightarrow X$ of $\mathsf{Set}^{\square^{op}}$.

Proposition

Let $X$ be equipped with the structure of a cubical Kan complex. Let $\sigma_{0}$ and $\sigma_{1}$ be $n$-cubes of $X$ which belong to $Z_{n}(X,x)$. Suppose that $\sigma_{0} \sim \sigma_{1}$. Then $\sigma_{1} \sim \sigma_{0}$.

Homotopy groups of a cubical Kan complex

Notation

We denote by $\pi_{n}(X,x)$ the set $Z_{n}(X,x) / \sim$.