# nLab Alexander-Whitney map

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

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.

###### Definition

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

$\Delta_{A,B} : C(A \otimes B) \to C(A) \otimes C(B)$

defined on two $n$-simplices $a \in A_n$ and $b \in B_n$ by

$\Delta_{A,B} : a \otimes b \mapsto \oplus_{p + q = n} (\tilde d^p a) \otimes (d^q_0 b) \,,$

where the front face map $\tilde d^p$ is that induced by

$[p] \to [p+q] : i \mapsto i$

and the back face $d^q_0$ map is that induced by

$[p] \to [p+q] : i \mapsto i+p \,.$
###### Definition

This AW map restricts to the normalized chains complex

$\Delta_{A,B} : N(A \otimes B) \to N(A) \otimes N(B) \,.$

## 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 $\nabla_{A,B}$:

$Id : N A \otimes N B \stackrel{\nabla_{A,B}}{\to} N(A \otimes B) \stackrel{\Delta_{A,B}}{\to} N A \otimes N B \,.$

The AW map is not symmetric.

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