nLab
cubical Kan complex

Contents

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 XX equipped with the following structure: for every integer n1n \geq 1, every integer 1in1 \leq i \leq n, every integer 0ϵ10 \leq \epsilon \leq 1, and every morphism f: n,i,ϵXf : \sqcap^{n,i,\epsilon} \rightarrow X of cubical sets, there is a morphism g: nXg : \square^{n} \rightarrow X of cubical sets such that the following diagram in Set op\mathsf{Set}^{\square^{op}} commutes.

n,i,ϵ i i,ϵ f n g X \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.

Last revised on March 16, 2020 at 20:54:32. See the history of this page for a list of all contributions to it.