nLab cubical geometric realisation

Idea

The category of cubical sets, just as with any reasonable category of shapes for higher structures, admits a geometric realisation functor to the category of topological spaces. The universal property of the category of cubes allows for a particularly economical and canonical construction of this functor compared to some other such categories of shapes.

Construction

We make use of the notation at category of cubes and cubical set, and denote the category of topological spaces by $\mathsf{Top}$. Let $I$ denote the topological unit interval, let $\bullet$ denote the topological point, let $0 : \bullet \rightarrow I$ and $1: \bullet \rightarrow I$ be the continuous map which pick out the endpoints $0$ and $1$ of $I$ respectively, and let $t : I \rightarrow \bullet$ be the canonical map.

Notation

We denote by $\left| - \right|_{\leq 1}$ the functor $\square_{\leq 1} \rightarrow \mathsf{Top}$ given by $I^{0} \mapsto \bullet$, $I^{1} \mapsto I$, $i_{0} \mapsto 0$, $i_{1} \mapsto 1$, and $p \mapsto t$.

Notation

We denote by $\left| - \right|_{\square}: \square \rightarrow \mathsf{Top}$ the canonical functor determined by $\left| - \right|_{\leq 1}$, the cartesian monoidal structure on $\mathsf{Top}$, and the universal property of $\square$.

Definition

The geometric realisation functor $\mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Top}$ is the canonical functor determined by $\left| - \right|_{\square}$, the fact that $\mathsf{Top}$ is co-complete, and the universal property of $\mathsf{Set}^{\square^{op}}$.

Related concepts

• topological realization

• cubulation