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homotopy groups of a cubical Kan complex

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A relation on the nn-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 XX be a cubical set, let x: 0Xx : \square^{0} \rightarrow X be a 00-cube of XX, and let n0n \geq 0 be an integer.

Notation

We denote by Z n(X,x)Z_{n}(X,x) the set of nn-cubes σ: nX\sigma : \square^{n} \rightarrow X of XX with the property that the following diagram in Set op\mathsf{Set}^{\square^{op}} commutes for every integer 1in1 \leq i \leq n and every integer 0ϵ10 \leq \epsilon \leq 1.

n1 p 0 y(I i1i ϵI ni) x n σ X \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)Z_{n}(X,x) given by identifying σ 0\sigma_{0} and σ 1\sigma_{1} if there is an (n+1)(n+1)-cube h: n+1Xh : \square^{n+1} \rightarrow X of XX such that the following diagrams in Set op\mathsf{Set}^{\square^{op}} commute

n ni 0 n+1 σ 0 h X \array{ \square^{n} & \overset{\square^{n} \otimes i_{0}}{\rightarrow} & \square^{n+1} \\ & \underset{\sigma_{0}}{\searrow} & \downarrow h \\ & & X }
n ni 1 n+1 σ 1 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 Set op\mathsf{Set}^{\square^{op}} commutes for every integer 1in1 \leq i \leq n and every integer 0ϵ10 \leq \epsilon \leq 1.

n p 0 y(I i1i ϵI n+1i) x n+1 σ X \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 hh, viewed as an arrow n 1X\square^{n} \otimes \square^{1} \rightarrow X of Set op\mathsf{Set}^{\square^{op}}, defines a homotopy? from σ 0: nX\sigma_{0} : \square^{n} \rightarrow X to σ 1: nX\sigma_{1} : \square^{n} \rightarrow X.

The relation \sim is in fact an equivalence relation

Proposition

Let σ\sigma be an nn-cube of XX which belongs to Z n(X,x)Z_{n}(X,x). Then σσ\sigma \sim \sigma.

Proof

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

Proposition

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

Homotopy groups of a cubical Kan complex

Notation

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

Last revised on April 10, 2018 at 20:03:19. See the history of this page for a list of all contributions to it.