Remark 10.2.2.19 (Face Operators). For every pair of integers $0 \leq i \leq n$, there is a unique increasing function $\delta _{n}^{i}: [n-1] \hookrightarrow [n]$ whose image is the set $[n] \setminus \{ i\} $, given concretely by the formula
If $X_{\bullet }$ is an augmented semisimplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$, then evaluation on the morphism $\delta ^{i}_{n}$ determines a map $d^{n}_{i}: X_{n} \rightarrow X_{n-1}$, which we will refer to as the $i$th face operator for the augmented semisimplicial object $X_{\bullet }$. If $n > 0$, this recover the face operators for the underlying semisimplicial object of $X_{\bullet }$ (Remark 10.2.2.5). In the case $n = 0$, we obtain a new operator $d^{0}_{0}: X_{0} \rightarrow X_{-1}$.