Proposition 10.2.6.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$. If $X_{\bullet }$ is split, then it is a colimit diagram in $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\overline{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \mathrm{min} }^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ be a splitting of $X_{\bullet }$, and let $C_{+}: \operatorname{{\bf \Delta }}_{+} \rightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}}$ denote the concatenation functor of Remark 10.2.6.2. Let us abuse notation by identifying $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} )$ with the cone $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )^{\triangleright }$. We wish to show that the augmented simplicial object
\[ (X_{\bullet } = \overline{X} \circ \operatorname{N}_{\bullet }( C_{+}^{\operatorname{op}} ) ): \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}} \]
is a colimit diagram in $\operatorname{\mathcal{C}}$.
Note that $[0]$ is initial when viewed as an object of the category $\operatorname{{\bf \Delta }}_{ \mathrm{min} }$, and therefore final when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )$. Unwinding the definitions, we see that the functor $\operatorname{N}_{\bullet }( C_{+}^{\operatorname{op}} )$ factors as a composition
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )^{\triangleright } \xrightarrow { \operatorname{N}_{\bullet }( C )^{\triangleright }} \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )^{\triangleright } \xrightarrow {R} \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min} }^{\operatorname{op}} ), \]
where $R$ is the identity when restricted to $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )$ and carries the cone point of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )^{\triangleright }$ to $[0]$. Applying Corollary 7.2.2.6, we deduce that $(\overline{X} \circ R): \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram. Consequently, to show that $X_{\bullet }$ is a colimit diagram, it will suffice to show that the functor $\operatorname{N}_{\bullet }( C^{\operatorname{op}} ): \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} )$ is right cofinal (Corollary 7.2.2.3). This is a special case of Corollary 7.2.3.7, since the concatenation functor $C$ is left adjoint to the inclusion $\operatorname{{\bf \Delta }}_{\mathrm{min}} \hookrightarrow \operatorname{{\bf \Delta }}$ (Remark 10.2.6.2). $\square$