# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Warning 2.5.8.10. Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. Then the unnormalized shuffle product $\bar{\triangledown }$ of Construction 2.5.7.3 induces a chain map $\overline{ \mathrm{EZ} }: \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B) \rightarrow \mathrm{C}_{\ast }(A \otimes B)$. However, the analogue of Proposition 2.5.8.9 for unnormalized Moore complexes is false: that is, the composite map

$\mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B) \xrightarrow { \overline{ \mathrm{EZ} } } \mathrm{C}_{\ast }(A \otimes B) \xrightarrow { \overline{\mathrm{AW}} } \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$

is usually not equal to the identity.