(also nonabelian homological algebra)
Denote by
$C \colon sAb \to Ch_\bullet^+$ the chains/Moore complex functor of the Dold-Kan correspondence;
$(sAb, \otimes)$ the monoidal category of simplicial abelian groups with tensor product given degreewise by the tensor product of abelian groups;
$(Ch_\bullet^+, \otimes)$ the monoidal category of chain complexes with its tensor product of chain complexes.
For $A,B \in sAb$ two simplicial abelian group, the Eilenberg-Zilber map (or Eilenberg-MacLane map or shuffle map) is the natural transformation of 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
$sgn(\mu,\nu)$ is the signature of the corresponding permutation,
the maps $s_{\mu}$ and $s_\nu$ are defined by:
and
The shift in the indices in Def. 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 sum in Def. may be understood as being over all non-degenerate simplices in the Cartesian product $\Delta[p] \times \Delta[q]$ of simplices. See at products of simplices for more on this.
This Eilenberg-Zilber map (Def. ) co/restricts to the normalized chain complex inside the Moore complex, to a chain map of the form:
The Eilenberg-Zilber map (Def. ) is a lax monoidal transformation that makes $C$ and $N$ into lax monoidal functors.
See at monoidal Dold-Kan correspondence for details.
For the next statement notice that both $sAb$ and $Ch_\bullet^+$ are in fact symmetric monoidal categories.
The EZ map (Def. ) 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/Alexander-Whitney deformation retraction)
Let
and denote
by $N(A), N(B) \,\in\, Ch^+_\bullet =$ ConnectiveChainComplexes their normalized chain complexes,
by $A \otimes B \,\in\, sAb$ the degreewise tensor product of abelian groups,
by $N(A) \otimes N(B)$ the tensor product of chain complexes.
Then there is a deformation retraction
where
$\nabla_{A,B}$ is the Eilenberg-Zilber map;
$\Delta_{A,B}$ is the Alexander-Whitney map.
For unnormalized chain complexes, where we have a homotopy equivalence, this is the original Eilenberg-Zilber theorem (Eilenberg & Zilber 1953, Eilenberg & MacLane 1954, Thm. 2.1). The above deformation retraction for normalized chain complexes is Eilenberg & MacLane 1954, Thm. 2.1a. Both are reviewed in May 1967, Cor. 29.10. Explicit description of the homotopy operator is given in Gonzalez-Diaz & Real 1999.
The Eilenberg-Zilber map induces a functor from simplicial Lie algebras to dg-Lie algebras (see here).
The Eilenberg-Zilber map controls the formula for transgression in group cohomology, see there fore more.
In the context of filtered spaces $X_*, Y_*$ and their associated fundamental crossed complexes $\Pi X_*, \Pi Y_*$ 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:
following the Eilenberg-Zilber theorem of
Review and further discussion:
Peter May, Section 29.7 of Simplicial objects in algebraic topology , Chicago Lectures in Mathematics, University of Chicago Press 1967 (ISBN:9780226511818, djvu, pdf)
Dan Quillen, part I, section 4 of: Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (jstor:1970725)
Jean-Louis Loday, Section 1.6 of: Cyclic Homology, Grund. Math. Wiss. 301, Springer, 1992 (doi:10.1007/978-3-662-21739-9)
A.P. Tonks, On the Eilenberg-Zilber Theorem for crossed complexes. J. Pure Appl. Algebra, 179~(1-2) (2003) 199–220,
Tim Porter, Section 11.2 of: Crossed Menagerie,
Kerodon, 2.5.7 The Shuffle Product (00RF)
The Eilenberg-Zilber Homomorphism (00RS)
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