Let be the chains/Moore complex functor of the Dold-Kan correspondence.
Let be the standard monoidal category structure given degreewise by the tensor product on Ab and let be the standard monoidal structure on the category of chain complexes.
For two abelian simplicial groups, the Alexander-Whitney map is the natural transformation on chain complexes
defined on two -simplices and by
where the front face map is that induced by
and the back face 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 and 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 :
The AW map is not symmetric.
Revised on November 4, 2010 14:25:08
by Urs Schreiber