Remark 10.2.2.5 (Face Operators). Let $X_{\bullet }$ be a semisimplicial object of an $\infty $-category. For every pair of integers $0 \leq i \leq n$ with $n > 0$, 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 the semisimplicial object $X_{\bullet }$.
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