Remark 10.2.6.23. More informally, the augmented decalage of a simplicial object $X_{\bullet }$ is given by the formula $\operatorname{Dec}_{+}(X)_{n} = X_{n+1}$. Moreover, for every pair of integers $0 \leq i \leq n$, the face and degeneracy operators
\[ d^{n}_{i}: \operatorname{Dec}_{+}(X)_{n} \rightarrow \operatorname{Dec}_{+}(X)_{n-1} \quad \quad s^{n}_{i}: \operatorname{Dec}_{+}(X)_{n} \rightarrow \operatorname{Dec}_{+}(X)_{n+1} \]
coincide with the face and degeneracy operators
\[ d^{n+1}_{i+1}: X_{n+1} \rightarrow X_{n} \quad \quad s^{n+1}_{i+1}: X_{n+1} \rightarrow X_{n+2}. \]