Proposition 10.1.3.23 (Existence of Skeleta). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 0$ be an integer. If $\operatorname{\mathcal{C}}$ admits pushouts, then every simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ admits an $n$-skeleton.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We will show that the functor
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n} )^{\operatorname{op}} \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}} \]
admits a left Kan extension $Y_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$; Corollary 7.3.6.13 then guarantees that there is an (essentially unique) morphism of simplicial objects $u: Y_{\bullet } \rightarrow X_{\bullet }$ which is the identity when restricted to $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n} )^{\operatorname{op}}$. By virtue of Corollary 7.3.5.8, it will suffice to show that for every integer $k > n$, the diagram
\[ F: \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}^{\leq n}_{ [k] / } )^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n})^{\operatorname{op}} \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}} \]
admits a colimit. Set $K = \{ 1, 2, \cdots , k\} $, let $P^{\leq n}(K)$ denote the collection of all subsets of $K$ having cardinality $\leq n$, and let $F_0$ denote the composition of $F$ with the right cofinal functor
\[ \operatorname{N}_{\bullet }( P^{\leq n}(K) ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{ [k] / } )^{\operatorname{op}} \]
supplied by Lemma 10.1.3.13. Let $Q \subseteq P^{\leq n}(K)$ denote the collection of nonempty subsets of $K$ of cardinality $\leq n$, so that $F_0$ can be identified with a functor $G: \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{\mathcal{C}}_{X_0/}$. Since $\operatorname{\mathcal{C}}$ admits pushouts, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{X_0/}$ admits finite limits (Example 7.6.3.28). In particular, the functor $G$ admits a colimit in $\operatorname{\mathcal{C}}_{X_0/}$, which we can identify with a colimit of $F_{0}$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (Remark 7.1.2.11). $\square$