Construction 10.2.5.18 (Face Cubes). Fix an integer $k \geq -1$, and let $\operatorname{\raise {0.1ex}{\square }}^{k+1}$ be the simplicial cube of dimension $k+1$ (Notation 2.4.5.2). In what follows, we will identify $\operatorname{\raise {0.1ex}{\square }}^{k+1}$ with the opposite of the nerve of the partially ordered set $P( [k] )$ of all subsets of $[k] = \{ 0 < 1 < \cdots < k \} $. Note that there is an isomorphism of categories $P( [k] ) \rightarrow (\operatorname{{\bf \Delta }}_{+,\operatorname{inj}})_{ / [k] }$, which carries each subset $J \subseteq [k]$ of cardinality $j+1$ to the unique strictly increasing function $[j] \hookrightarrow [k]$ having image $J$. If $C_{\bullet }$ is an augmented semisimplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$, we let $\tau _{k}: \operatorname{\raise {0.1ex}{\square }}^{k+1} \rightarrow \operatorname{\mathcal{C}}$ denote the composite functor
We will refer to $\tau _{k}$ as the $k$th face cube of the augmented semisimplicial object $C_{\bullet }$.