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Construction (Face Operators). Let $n$ be a positive integer. For $0 \leq i \leq n$, we let $\delta ^{i}_{n}: [n-1] \rightarrow [n]$ denote the unique strictly increasing function whose image does not contain the element $i$, given concretely by the formula

\[ \delta ^{i}_{n}( j) = \begin{cases} j & \text{ if } j < i \\ j+1 & \text{ if } j \geq i. \end{cases} \]

If $C_{\bullet }$ is a (semi)simplicial object of a category $\operatorname{\mathcal{C}}$, then we can evaluate $C_{\bullet }$ on the morphism $\delta ^{i}_{n}$ to obtain a morphism from $C_{n}$ to $C_{n-1}$. We will denote this morphism by $d^{n}_{i}: C_{n} \rightarrow C_{n-1}$ and refer to it as the $i$th face operator.