Construction 10.2.6.22 (Decalage). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. We let $\operatorname{Dec}_{+}( X )_{\bullet }$ denote the augmented simplicial object of $\operatorname{\mathcal{C}}$ given by the composition
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \xrightarrow { \operatorname{N}_{\bullet }(C^{\operatorname{op}}_{+}) } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}, \]
where $C_{+}$ denote the concatenation functor of Remark 10.2.6.2. We will refer to $\operatorname{Dec}_{+}(X)_{\bullet }$ as the augmented decalage of $X_{\bullet }$. We let $\operatorname{Dec}(X)_{\bullet }$ denote the underlying simplicial object of $\operatorname{Dec}_{+}(X)_{\bullet }$, given by the composition
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { \operatorname{N}_{\bullet }(C^{\operatorname{op}}) } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}. \]
We will defer to $\operatorname{Dec}(X)_{\bullet }$ as the decalage of $\operatorname{Dec}(X)_{\bullet }$.