Kerodon

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Notation 1.1.1.9. For every pair of integers $0 \leq i \leq n$ we let $\sigma ^{i}: [n+1] \rightarrow [n]$ denote the function given by the formula

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

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

Dually, if $C^{\bullet }$ is a cosimplicial object of a category $\operatorname{\mathcal{C}}$, then the evaluation on $C^{\bullet }$ on the morphism $\sigma ^{i}$ determines a map $s^{i}: C^{n+1} \rightarrow C^{n}$, which we refer to as the $i$th codegeneracy map.