Remark 10.2.6.25. Let $X_{\bullet }$ be a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. Then the augmented simplicial object $\operatorname{Dec}_{+}(X)_{\bullet }$ is split: it admits a splitting given by the diagram
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}. \]
In particular, Proposition 10.2.6.9 guarantees that $\operatorname{Dec}_{+}(X)_{\bullet }$ is a colimit diagram in $\operatorname{\mathcal{C}}$: that is, it exhibits the object $X_{0} \in \operatorname{\mathcal{C}}$ as a geometric realization of the decalage $\operatorname{Dec}(X)_{\bullet }$.