and
nonabelian homological algebra
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.
For $A,B \in sAb$ two abelian simplicial groups, the Eilenberg-Zilber map or Eilenberg-MacLane map or shuffle map is the natural transformation on chain complexes
defined on two $n$-simplices $a \in A_p$ and $b \in B_q$ by
where the sum is over all $(p,q)$-shuffles
and the corresponding degeneracy maps are
and
(The shift in the indices is to be coherent with the convention that the shuffle $(\mu, \nu)$ is a permutation of $\{1, \dots, p+q\}$. In many references the shift disappears by making it a permutation of $\{0, \dots, p+q-1\}$ instead.) The sign $sign(\mu,\nu) \in \{-1,1\}$ is the signature of the corresponding permutation.
The sum may be understood as being over all non-degenerate simplices in the product $\Delta[p] \times \Delta[q]$. See products of simplices for more on this.
This map restricts to the normalized chains complex
The specific maps introduced by Eilenberg-Mac Lane have stronger properties which for simplicial sets $K,L$ make $C(K) \otimes C(L)$ a strong deformation retract of $C(K \times L)$. This is exploited in
which has been a foundation of the subject of Homological Perturbation Theory. The homotopies play a key role in the formulae and calculations.
The Eilenberg-Zilber map is a lax monoidal transformation that makes $C$ and $N$ into lax monoidal functors.
See monoidal Dold-Kan correspondence for details.
On normalized chain complexes the EZ map has a left inverse, given by the Alexander-Whitney map $\Delta_{A,B}$:
For all $X,Y$ the EZ map $\nabla_{X,Y}$ is a quasi-isomorphism and in fact a chain homotopy equivalence.
This is in 29.10 of (May).
For the next statement notice that both $sAb$ and $Ch_\bullet^+$ are in fact symmetric monoidal categories.
The EZ map is symmetric in that for all $A,B \in sAb$ the square
commutes, where $\sigma$ denotes the symmetry isomorphism in $sAb$ and $Ch_\bullet^+$.
Eilenberg-Zilber map
In the context of filtered spaces $X_*, Y_*$ and their associated fundamental crossed complex?es $\Pi X_*, \Pi Y_*$ there is a natural Eilenberg-Zilber morphism
which is difficult to define directly because of the complications of the tensor product of crossed complexes, but has a direct definition in terms of the associated cubical homotopy $\omega$–groupoids. This morphism is an isomorphism of free crossed complexes if $X_*, Y_*$ are the skeletal filtrations of CW-complexes. For more on all this, see the book Nonabelian Algebraic Topology p. 533.
The Eilenberg-Zilber map was introduced in (5.3) of
See also 29.7 of
Peter May, Simplicial objects in algebraic topology , Chicago Lectures in Mathematics, Chicago, (1967) (djvu) ,
A.P. Tonks, On the Eilenberg-Zilber Theorem for crossed complexes. J. Pure Appl. Algebra, 179~(1-2) (2003) 199–220,
section 11.2 of
or section 1.6 of