Variant 10.2.6.7. In the situation of Example 10.2.6.6, let $\mathrm{N}_{\ast }^{\mathrm{aug}}(A)$ denote the augmented normalized Moore complex of $A_{\bullet }$ (Variant 10.2.2.22). It follows from (10.14) that, for every integer $n \geq 0$, the operator
\[ \mathrm{C}^{\operatorname{aug}}_{n}(A) = A_{n} \xrightarrow {h_ n} A_{n+1} = \mathrm{C}^{\operatorname{aug}}_{n+1}(A) \]
carries degenerate $n$-simplices of $A_{\bullet }$ to degenerate $(n+1)$-simplices of $A_{\bullet }$, and therefore descends to an operator $\overline{h}_{n}: \mathrm{N}^{\operatorname{aug}}_{n}(A) \rightarrow \mathrm{N}^{\operatorname{aug}}_{n+1}(A)$. The collection of homomorphisms $\{ \overline{h}_{n} \} $ then determine a contracting homotopy for the chain complex $\mathrm{N}_{\ast }^{\mathrm{aug}}(A)$.