Notation 10.2.0.4 (Face and Degeneracy Operators). Let $X_{\bullet }$ be a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. For every pair of integers $0 \leq i \leq n$, we let $s^{n}_{i}: X_{n} \rightarrow X_{n+1}$ denote the morphism induced by the surjection $\sigma ^{i}_{n}: [n+1] \twoheadrightarrow [n]$ of Construction 1.1.2.1; we will refer to $s^{n}_{i}$ as the *$i$th degeneracy operator* for the simplicial object $X_{\bullet }$. If $n > 0$, we let $d^{n}_{i}: X_{n} \rightarrow X_{n-1}$ denote the morphism induced by the inclusion of linearly ordered sets $\delta ^{i}_{n}: [n-1] \hookrightarrow [n]$ introduced in Construction 1.1.1.4. We will refer to $d^{n}_{i}$ as the *$i$th face operator* of $X_{\bullet }$.

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