# Kerodon

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Proposition 10.1.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$, and let $n$ be an integer. The following conditions are equivalent:

$(1)$

The simplicial object $X_{\bullet }$ is $n$-skeletal, in the sense of Definition 10.1.3.4.

$(2)$

Let $k \geq 0$ be a nonnegative integer and set $K = \{ 1, 2, \cdots , k \}$. Then the $k$th degeneracy cube

$\sigma _{k}: \operatorname{\raise {0.1ex}{\square }}^{k} = \operatorname{N}_{\bullet }( P(K) ) \rightarrow \operatorname{\mathcal{C}}$

exhibits $X_{k}$ as a colimit of the diagram $\sigma _{k} |_{ \operatorname{N}_{\bullet }( P^{\leq n}(K) ) }$.

$(3)$

For every nonnegative integer $k > n$, the $k$-degeneracy cube $\sigma _{k}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows immediately from Lemma 10.1.3.13 (together with Corollary 7.2.2.3). For each integer $k \geq 0$, set $K = \{ 1, 2, \cdots , k \}$ and consider the following conditions:

$(2_ k)$

The degeneracy cube $\sigma _{k}: \operatorname{N}_{\bullet }( P(K) ) \rightarrow \operatorname{\mathcal{C}}$ exhibits $X_{k}$ as a colimit of the diagram $\sigma _{k}|_{ \operatorname{N}_{\bullet }( P^{\leq n}(K) ) }$.

$(3_ k)$

The degeneracy cube $\sigma _{k}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.

Note that condition $(2_ k)$ is automatic for $k \leq n$. We will complete the proof by showing that if $k > n$ and condition $(2_{\ell } )$ is satisfied for every integer $0 \leq \ell < k$, then conditions $(2_ k)$ and $(3_ k)$ are equivalent. Our hypothesis that condition $(2_{\ell } )$ is satisfied for $\ell < k$ guarantees that the functor $\sigma _{k} |_{ \operatorname{N}_{\bullet }( P^{\leq k-1}(K) ) }$ is left Kan extended from the full subcategory $\operatorname{N}_{\bullet }( P^{\leq n}(K) ) \subseteq \operatorname{N}_{\bullet }( P^{\leq k-1}(K) )$. The equivalence of $(2_ k)$ and $(3_ k)$ is therefore a special case of Corollary 7.3.8.2. $\square$