Construction 10.2.3.10 (Degeneracy Cubes). Fix an integer $k \geq 0$, set $K = \{ 1, 2, \cdots , k \} $, and let $\operatorname{\raise {0.1ex}{\square }}^{k} = \operatorname{\raise {0.1ex}{\square }}^{K}$ denote the simplicial cube of dimension $k$ (Notation 2.4.5.2). Recall that $\operatorname{\raise {0.1ex}{\square }}^{k}$ can be identified with the nerve of the set $P(K)$ of subsets of $K$, partially ordered with respect to inclusion.
For every subset $J \subseteq K$ having cardinality $j$, let $\alpha _{J}: [k] \twoheadrightarrow [j]$ denote the nondecreasing function function which carries an element $k' \in [k]$ to the cardinality of the intersection $J \cap \{ 1,2, \cdots , k' \} $. The construction $J \mapsto \alpha _{J}$ then determines a functor from $P(K)^{\operatorname{op}}$ to the coslice category $\operatorname{{\bf \Delta }}_{[k] / }$.
If $\operatorname{\mathcal{C}}$ is an $\infty $-category and $X_{\bullet }$ is a simplicial object of $\operatorname{\mathcal{C}}$, we let $\sigma _{k}: \operatorname{\raise {0.1ex}{\square }}^{k} \rightarrow \operatorname{\mathcal{C}}$ denote the map given by the composition
We will refer to $\sigma _{k}$ as the $k$th degeneracy cube of the simplicial object $X_{\bullet }$.