nLab cubical Kan complex

Definition

We shall make use of notions introduced at cubical set, and of the notation of this page.

Definition

A cubical Kan complex is a cubical set $X$ equipped with the following structure: for every integer $n \geq 1$, every integer $1 \leq i \leq n$, every integer $0 \leq \epsilon \leq 1$, and every morphism $f : \sqcap^{n,i,\epsilon} \rightarrow X$ of cubical sets, there is a morphism $g : \square^{n} \rightarrow X$ of cubical sets such that the following diagram in $\mathsf{Set}^{\square^{op}}$ commutes.

$$\array{ \sqcap^{n,i,\epsilon} & & \\ \mathllap{i_{i,\epsilon}} \downarrow & \overset{f}{\searrow} & \\ \square^{n} & \underset{g}{\rightarrow} & X }$$

Homotopy groups

Cubical Kan complexes admit a notion of homotopy group. The theory of these homotopy groups can be developed analogously to the theory of homotopy groups of a topological space. See homotopy groups of a cubical Kan complex.