Construction 2.5.5.7 (The Normalized Moore Complex: First Construction). Let $A_{\bullet }$ be a simplicial abelian group. We let $\mathrm{N}_{\ast }(A)$ denote the chain complex given by the quotient $\mathrm{C}_{\ast }(A) / \mathrm{D}_{\ast }(A)$, where $\mathrm{C}_{\ast }(A)$ is the Moore complex of Construction 2.5.5.1 and $\mathrm{D}_{\ast }(A) \subseteq \mathrm{C}_{\ast }(A)$ is the subcomplex of Proposition 2.5.5.6. We will refer to $\mathrm{N}_{\ast }(A)$ as the normalized Moore complex of the simplicial abelian group $A_{\bullet }$.
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