Proposition 10.2.4.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be an integer, and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. Then $X_{\bullet }$ is $n$-coskeletal (in the sense of Definition 10.2.4.9) if and only if its underlying semisimplicial object is $n$-coskeletal (in the sense of Variant 10.2.4.14).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix an integer $k \geq 0$, $\operatorname{{\bf \Delta }}_{ / [k] }^{\leq n}$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{ / [k] }$ spanned by those objects which correspond to nondecreasing functions $\alpha : [m] \rightarrow [k]$ where $m \leq n$, and let $\operatorname{\mathcal{J}}$ be the full subcategory of $\operatorname{{\bf \Delta }}_{ / [k]}^{\leq n}$ spanned by those objects where $\alpha $ is strictly increasing. Unwinding the definitions, we see that $X_{\bullet }$ is $n$-coskeletal if and only if, for every integer $k \geq 0$, the composite functor
\[ F: \operatorname{N}_{\bullet }( (\operatorname{{\bf \Delta }}_{ / [k] }^{\leq n}))^{\triangleright } \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{ / [k] }^{\triangleright } ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}^{\operatorname{op}} \]
is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Similarly, the underlying semisimplicial object of $X_{\bullet }$ is $n$-coskeletal if and only if, for every integer $k \geq 0$, the restriction $F|_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{J}})^{\triangleright } }$ is a colimit diagram in $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Consequently, to show that these conditions are equivalent, it will suffice to prove that the inclusion functor $\operatorname{N}_{\bullet }(\operatorname{\mathcal{J}}) \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{ / [k] }^{\leq n})$ is right cofinal (Corollary 7.2.2.3). This is a special case of Corollary 7.2.3.7, since the inclusion functor $\operatorname{\mathcal{J}}\hookrightarrow \operatorname{{\bf \Delta }}_{ / [k] }^{\leq n}$ has a left adjoint (which carries a nondecreasing function $\alpha : [m] \rightarrow [k]$ to the inclusion map $\operatorname{im}(\alpha ) \hookrightarrow [k]$; here we abuse notation by identifying $\operatorname{im}(\alpha )$ with the corresponding object of $\operatorname{{\bf \Delta }}$). $\square$