nLab
cubical geometric realisation

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 Top\mathsf{Top}. Let II denote the topological unit interval, let \bullet denote the topological point, let 0:I0 : \bullet \rightarrow I and 1:I1: \bullet \rightarrow I be the continuous map which pick out the endpoints 00 and 11 of II respectively, and let t:It : I \rightarrow \bullet be the canonical map.

Notation

We denote by || 1\left| - \right|_{\leq 1} the functor 1Top\square_{\leq 1} \rightarrow \mathsf{Top} given by I 0I^{0} \mapsto \bullet, I 1II^{1} \mapsto I, i 00i_{0} \mapsto 0, i 11i_{1} \mapsto 1, and ptp \mapsto t.

Notation

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

Definition

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