Warning 10.2.6.8. Let $A_{\bullet }$ be an augmented simplicial abelian group. In general, not every contracting homotopy for the chain complex $\mathrm{N}_{\ast }^{\mathrm{aug}}(A)$ can be obtained from the construction of Variant 10.2.6.7. A splitting of $A_{\bullet }$ determines a system of homomorphisms $\{ h_{n}: A_{n} \rightarrow A_{n+1} \} _{n \geq 0}$ which satisfy the identity $h_{n+1} \circ h_{n} = s^{n+1}_{0} \circ h_{n}$ (Remark 10.2.6.4). In particular, the composition $h_{n+1} \circ h_{n}$ carries every $n$-simplex of $A_{\bullet }$ to a degenerate $(n+2)$-simplex of $A_{\bullet }$. It follows that the composite map
\[ \mathrm{N}_{n}^{\operatorname{aug}}( A ) \xrightarrow { \overline{h}_{n} } \mathrm{N}_{n+1}^{\operatorname{aug}}( A ) \xrightarrow { \overline{h}_{n+1} } \mathrm{N}_{n+2}^{\operatorname{aug}}(A) \]
vanishes; for a general contracting homotopy, the analogous statement need not be true.