Definition 10.2.6.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$ (Definition 10.2.1.12). A splitting of $X_{\bullet }$ is a functor $\overline{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \mathrm{min}}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ for which the composition
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \xrightarrow {C^{\operatorname{op}}_{+}} \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \xrightarrow { \overline{X} } \operatorname{\mathcal{C}} \]
is equal to $X_{\bullet }$; here $C_{+}$ denotes the concatenation functor $[n] \mapsto [n] \star [0]$ of Remark 10.2.6.2. We will say that the augmented simplicial object $X_{\bullet }$ is split if there exists a splitting of $X_{\bullet }$.