# nLab Alexander-Whitney map

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

Let $C:\mathrm{sAb}\to {\mathrm{Ch}}_{•}^{+}$ be the chains/Moore complex functor of the Dold-Kan correspondence.

Let $\left(\mathrm{sAb},\otimes \right)$ be the standard monoidal category structure given degreewise by the tensor product on Ab and let $\left({\mathrm{Ch}}_{•}^{+},\otimes \right)$ be the standard monoidal structure on the category of chain complexes.

###### Definition

For $A,B\in \mathrm{sAb}$ two abelian simplicial groups, the Alexander-Whitney map is the natural transformation on chain complexes

${\Delta }_{A,B}:C\left(A\otimes B\right)\to C\left(A\right)\otimes C\left(B\right)$\Delta_{A,B} : C(A \otimes B) \to C(A) \otimes C(B)

defined on two $n$-simplices $a\in {A}_{n}$ and $b\in {B}_{n}$ by

${\Delta }_{A,B}:a\otimes b↦{\oplus }_{p+q=n}\left({\stackrel{˜}{d}}^{p}a\right)\otimes \left({d}_{0}^{q}b\right)\phantom{\rule{thinmathspace}{0ex}},$\Delta_{A,B} : a \otimes b \mapsto \oplus_{p + q = n} (\tilde d^p a) \otimes (d^q_0 b) \,,

where the front face map ${\stackrel{˜}{d}}^{p}$ is that induced by

$\left[p\right]\to \left[p+q\right]:i↦i$[p] \to [p+q] : i \mapsto i

and the back face ${d}_{0}^{q}$ map is that induced by

$\left[p\right]\to \left[p+q\right]:i↦i+p\phantom{\rule{thinmathspace}{0ex}}.$[p] \to [p+q] : i \mapsto i+p \,.
###### Definition

This AW map restricts to the normalized chains complex

${\Delta }_{A,B}:N\left(A\otimes B\right)\to N\left(A\right)\otimes N\left(B\right)\phantom{\rule{thinmathspace}{0ex}}.$\Delta_{A,B} : N(A \otimes B) \to N(A) \otimes N(B) \,.

## Properties

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 ${\nabla }_{A,B}$:

$\mathrm{Id}:NA\otimes NB\stackrel{{\nabla }_{A,B}}{\to }N\left(A\otimes B\right)\stackrel{{\Delta }_{A,B}}{\to }NA\otimes NB\phantom{\rule{thinmathspace}{0ex}}.$Id : N A \otimes N B \stackrel{\nabla_{A,B}}{\to} N(A \otimes B) \stackrel{\Delta_{A,B}}{\to} N A \otimes N B \,.

The AW map is not symmetric.

Revised on November 4, 2010 14:25:08 by Urs Schreiber (87.212.203.135)