# Kerodon

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Construction 10.1.3.10 (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

$\operatorname{\raise {0.1ex}{\square }}^{k} \simeq \operatorname{N}_{\bullet }( P(K) ) \xrightarrow {J \mapsto \alpha _{J} } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{[k]/} )^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}.$

We will refer to $\sigma _{k}$ as the $k$th degeneracy cube of the simplicial object $X_{\bullet }$.