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Alexander-Whitney map

Definition

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

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

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

$${\Delta }_{A,B}:a\otimes b\mapsto {\oplus }_{p+q=n}({\tilde{d}}^{p}a)\otimes (d_0^{q}b),$$\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

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

and the back face $d_0^q$ map is that induced by

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

Definition

This AW map restricts to the normalized chains complex

$${\Delta }_{A,B}:N(A\otimes B)\to N(A)\otimes N(B).$$\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 ${\nabla }_{A,B}$:

$$\mathrm{Id}:NA\otimes NB\stackrel{{\nabla }_{A,B}}{\to }N(A\otimes B)\stackrel{{\Delta }_{A,B}}{\to }NA\otimes NB.$$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.

Related concepts

• Alexander-Whitney map

• Eilenberg-Zilber map