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 $\mathsf{Set}^{\square^{op}}$.

Created on January 26, 2021 at 19:25:42.
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