$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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

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

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

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