# Contents

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

Created on January 26, 2021 at 19:25:42. See the history of this page for a list of all contributions to it.