Variant 10.2.2.22. Let $A_{\bullet }$ be an augmented simplicial object of the category of abelian groups. Let us abuse notation by identifying $A_{\bullet }$ with the underlying simplicial abelian group, and let
\[ \mathrm{D}_{\ast }(A) \subseteq \mathrm{C}_{\ast }(A) \subseteq \mathrm{C}_{\ast }^{\operatorname{aug}}( A ) \]
be the subcomplex generated by the images of the degeneracy operators (see Proposition 2.5.5.6). We let $\mathrm{N}_{\ast }^{\operatorname{aug}}(A)$ denote the quotient complex $\mathrm{C}^{\operatorname{aug}}_{\ast }(A) / \mathrm{D}_{\ast }(A)$, which we will refer to as the normalized augmented Moore complex of $A_{\bullet }$. Note that, when restricted to nonnegative degrees, this recovers the normalized Moore complex of the underlying simplicial abelian group (Construction 2.5.5.7).