and
nonabelian homological algebra
Let $C : sAb \to Ch_\bullet^+$ be the chains/Moore complex functor of the Dold-Kan correspondence.
Let $(sAb, \otimes)$ be the standard monoidal category structure given degreewise by the tensor product on Ab and let $(Ch_\bullet^+, \otimes)$ be the standard monoidal structure on the category of chain complexes.
For $A,B \in sAb$ two abelian simplicial groups, the Alexander-Whitney map is the natural transformation on chain complexes
defined on two $n$-simplices $a \in A_n$ and $b \in B_n$ by
where the front face map $\tilde d^p$ is that induced by
and the back face $d^q_0$ map is that induced by
This AW map restricts to the normalized chains complex
The Alexander-Whitney map is an oplax monoidal transformation that makes $C$ and $N$ into oplax monoidal functors. For details see monoidal Dold-Kan correspondence.
On normalized chain complexes the AW map has a right inverse, given by the Eilenberg-Zilber map $\nabla_{A,B}$:
The AW map is not symmetric.
Alexander-Whitney map