Remark 10.2.2.21 (Augmented Moore Complexes). Let $A_{\bullet }$ be an augmented semisimplicial object of the category of abelian groups. For each $n \geq 0$, let $\partial : A_{n} \rightarrow A_{n-1}$ denote the group homomorphism given by the alternating sum
\[ \partial (\sigma ) = \sum _{i = 0}^{n} (-1)^{i} d^{n}_ i(\sigma ). \]
The diagram
\[ \cdots \rightarrow A_{2} \xrightarrow {\partial } A_{1} \xrightarrow {\partial } A_{0} \xrightarrow {\partial } A_{-1} \]
is a chain complex of abelian groups which we will denote by $\mathrm{C}^{\operatorname{aug}}_{\ast }(A)$ and refer to as the augmented Moore complex of $A_{\bullet }$. Note that, when restricted to nonnegative degrees, this recovers the Moore complex of the underlying semisimplicial abelian group (see Construction 2.5.5.1).