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Construction (Degeneracy Operators). For every pair of integers $0 \leq i \leq n$ we let $\sigma ^{i}_{n}: [n+1] \twoheadrightarrow [n]$ denote the nondecreasing function given by the formula

\[ \sigma ^{i}_{n}( j) = \begin{cases} j & \text{ if } j \leq i \\ j-1 & \text{ if } j > i. \end{cases} \]

If $C_{\bullet }$ is a simplicial object of a category $\operatorname{\mathcal{C}}$, then we can evaluate $C_{\bullet }$ on the morphism $\sigma ^{i}_{n}$ to obtain a morphism from $C_{n}$ to $C_{n+1}$. We will denote this map by $s^{n}_{i}: C_{n} \rightarrow C_{n+1}$ and refer to it as the $i$th degeneracy operator.