Construction 2.5.5.1 (The Moore Complex). Let $A_{\bullet }$ be a semisimplicial abelian group (Variant 1.1.1.6). For each $n \geq 1$, we define a group homomorphism $\partial : A_{n} \rightarrow A_{n-1}$ by the formula
where $d_{i}: A_{n} \rightarrow A_{n-1}$ is the $i$th face map (Notation 1.1.1.8). For $n \geq 2$ and $\sigma \in A_{n}$, we compute
where the final equality follows from the identity $d_{i} \circ d_{j} = d_{j-1} \circ d_{i}$ for $0 \leq i < j \leq n$ (see Exercise 1.1.1.10). We let $\mathrm{C}_{\ast }(A)$ denote the chain complex of abelian groups given by
where the differential is given by $\partial $. We will refer to $\mathrm{C}_{\ast }(A)$ as the Moore complex of the semisimplicial abelian group $A_{\bullet }$.
If $A_{\bullet }$ is a simplicial abelian group, we let $\mathrm{C}_{\ast }(A)$ denote the Moore complex of the semisimplicial abelian group underlying $A_{\bullet }$ (Remark 1.1.1.7).