nLab
Alexander-Whitney map

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Homological algebra

homological algebra

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nonabelian homological algebra

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Definition

Let C:sAbCh +C : sAb \to Ch_\bullet^+ be the chains/Moore complex functor of the Dold-Kan correspondence.

Let (sAb,)(sAb, \otimes) be the standard monoidal category structure given degreewise by the tensor product on Ab and let (Ch +,)(Ch_\bullet^+, \otimes) be the standard monoidal structure on the category of chain complexes.

Definition

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

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

defined on two nn-simplices aA na \in A_n and bB nb \in B_n by

Δ A,B:ab p+q=n(d˜ pa)(d 0 qb), \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 d˜ p\tilde d^p is that induced by

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

and the back face d 0 qd^q_0 map is that induced by

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

This AW map restricts to the normalized chains complex

Δ A,B:N(AB)N(A)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 CC and NN 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 A,B\nabla_{A,B}:

Id:NANB A,BN(AB)Δ A,BNANB. 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)