Example 10.2.6.6. Let $A_{\bullet }$ be an augmented simplicial abelian group, and let
\[ \mathrm{C}_{\ast }^{\mathrm{aug}}( A ) = ( \cdots \rightarrow A_{2} \xrightarrow {\partial } A_{1} \xrightarrow {\partial } A_0 \xrightarrow {\partial } A_{-1} ) \]
denote its augmented Moore complex (Remark 10.2.2.21). Suppose we are given a splitting of $A_{\bullet }$, and let $\{ h_{n}: A_{n} \rightarrow A_{n+1} \} _{n \geq -1}$ be the extra degeneracy operators described in Remark 10.2.6.4. Then the collection $\{ h_{n} \} $ is a contracting homotopy for $\mathrm{C}_{\ast }^{\operatorname{aug}}( A )$, in the sense of Definition 2.5.0.5: that is, the homomorphism
\[ ( h_{n-1} \circ \partial + \partial \circ h_{n} ): A_{n} \rightarrow A_{n} \]
is equal to the identity for each $n \geq -1$ (where we adopt the convention that $h_{n} \circ \partial = 0$ for $n = -1$). This follows from the calculation
\begin{eqnarray*} h_{n-1} \circ \partial + \partial \circ h_{n} & = & (\sum _{i=0}^{n} (-1)^{i} h_{n-1} \circ d^{n}_{i}) + (\sum _{j=0}^{n+1} (-1)^{j} d^{n+1}_{j} \circ h_{n}) \\ & = & (\sum _{i=0}^{n} (-1)^{i} (h_{n-1} \circ d^{n}_{i} - d^{n+1}_{i+1} \circ h_{n} ) ) + d^{n+1}_{0} \circ h_{n} \\ & = & \operatorname{id}_{ A_{n} }. \end{eqnarray*}
where the final equality follows from the identities (10.11).