Proposition 10.2.2.29. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite coproducts and let $X_{\bullet }$ be a semisimplicial object of $\operatorname{\mathcal{C}}$. Then there exists a simplicial object $X_{\bullet }^{+}$ of $\operatorname{\mathcal{C}}$ and a natural transformation of semisimplicial objects $f_{\bullet }: X_{\bullet } \rightarrow X_{\bullet }^{+}$ which exhibits $X_{\bullet }^{+}$ as a left Kan extension of $X_{\bullet }$ along the inclusion map $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We will show that $X_{\bullet }$ satisfies the criterion of Proposition 7.3.5.1. Fix an object $[n] \in \operatorname{{\bf \Delta }}$, let $\operatorname{\mathcal{E}}$ denote the fiber product $\operatorname{{\bf \Delta }}_{\operatorname{inj}} \times _{ \operatorname{{\bf \Delta }}} \operatorname{{\bf \Delta }}_{ [n] / }$, and let $F$ denote the composite map
\[ \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}^{\operatorname{op}}) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}. \]
We wish to show that $F$ admits a colimit in $\operatorname{\mathcal{C}}$.
By definition, objects of the category $\operatorname{\mathcal{E}}$ can be identified with pairs $([m], \alpha )$, where $[m]$ is an object of $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ is an integer and $\alpha : [n] \rightarrow [m]$ is a nondecreasing function. Let $\operatorname{\mathcal{E}}_0 \subseteq \operatorname{\mathcal{E}}$ denote the full subcategory spanned by those objects $([m], \alpha )$ where $\alpha $ is a surjection. Note that any morphism $\alpha : [n] \rightarrow [m]$ in $\operatorname{{\bf \Delta }}$ factors uniquely as a composition $[n] \xrightarrow {\alpha '} [m'] \xrightarrow {\beta } [m]$, where $\alpha '$ is a surjection and $\beta $ is an injection. The pair $([m'], \alpha ')$ is then an object of the subcategory $\operatorname{\mathcal{E}}_0$, and the morphism $\beta : ([m'], \alpha ') \rightarrow ([m], \alpha )$ exhibits exhibits $([m'], \alpha ')$ as a $\operatorname{\mathcal{E}}_0$-coreflection of $([m], \alpha )$ (see Definition 6.2.2.1). It follows that the inclusion map $\operatorname{N}_{\bullet }( \operatorname{\mathcal{E}}^{\operatorname{op}}_0 ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{E}}^{\operatorname{op}} )$ is right cofinal (Corollary 7.2.2.7). Consequently, to show that $F$ admits a colimit in $\operatorname{\mathcal{C}}$, it will suffice to show that the restriction $F|_{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{E}}^{\operatorname{op}}_{0} ) }$ admits a colimit in $\operatorname{\mathcal{C}}$ (Corollary 7.2.2.10). Since the category $\operatorname{\mathcal{E}}_0$ has finitely many objects and only identity morphisms, this follows from our assumption that $\operatorname{\mathcal{C}}$ admits finite coproducts. $\square$