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\begin{document}

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\section*{Eilenberg-Zilber map}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{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]].

\begin{defn}
\label{}\hypertarget{}{}
For $A,B \in sAb$ two abelian [[simplicial group]]s, the \textbf{Eilenberg-Zilber map} or \textbf{Eilenberg-MacLane map} or \textbf{shuffle map} is the [[natural transformation]] on [[chain complex]]es

\begin{displaymath}
\nabla_{A,B} :  C(A) \otimes C(B) \to C(A \otimes B)
\end{displaymath}
defined on two $n$-simplices $a \in A_p$ and $b \in B_q$ by

\begin{displaymath}
\nabla_{A,B} : a \otimes b \mapsto 
   \sum_{(\mu,\nu)} sign(\mu,\nu) (s_\nu(a)) \otimes (s_\mu(b))
   \;\;
   \in
   C_{p+q}(A \otimes B)
   =
   A_{p+q} \otimes B_{p+q}
  \,,
\end{displaymath}
where the sum is over all $(p,q)$-[[shuffle]]s

\begin{displaymath}
(\mu,\nu) = (\mu_1, \cdots, \mu_p, \nu_1, \cdots, \nu_q)
\end{displaymath}
and the corresponding degeneracy maps are

\begin{displaymath}
s_{\mu} = s_{\mu_p - 1} \circ \cdots s_{\mu_2 - 1} \circ s_{\mu_1 - 1}
\end{displaymath}
and

\begin{displaymath}
s_{\nu} = s_{\nu_q - 1} \circ \cdots s_{\nu_2 - 1} \circ s_{\nu_1 - 1}
  \,.
\end{displaymath}
(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 a permutation|signature]] of the corresponding [[permutation]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
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.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
This map restricts to the normalized chain complex

\begin{displaymath}
\nabla_{A,B} :  N(A) \otimes N(B) \to N(A \otimes B) 
  \,.
\end{displaymath}
\end{prop}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{}\hypertarget{}{}
The Eilenberg-Zilber map is a [[lax monoidal transformation]] that makes $C$ and $N$ into [[lax monoidal functor]]s.

\end{prop}
See [[monoidal Dold-Kan correspondence]] for details.

\begin{prop}
\label{}\hypertarget{}{}
On normalized chain complexes the EZ map has a [[left inverse]], given by the [[Alexander-Whitney map]] $\Delta_{A,B}$:

\begin{displaymath}
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
  \,.
\end{displaymath}
\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
For all $X,Y$ the EZ map $\nabla_{X,Y}$ is a [[quasi-isomorphism]] and in fact a chain [[homotopy equivalence]].

\end{prop}
This is in 29.10 of (\hyperlink{May}{May}).

For the next statement notice that both $sAb$ and $Ch_\bullet^+$ are in fact [[symmetric monoidal categories]].

\begin{prop}
\label{}\hypertarget{}{}
The EZ map is [[symmetric monoidal functor|symmetric]] in that for all $A,B \in sAb$ the square

\begin{displaymath}
\itexarray{
    C A \otimes C B &\stackrel{\sigma}{\to}& C B \otimes C A
    \\
    {}^{\mathllap{\nabla_{A,B}}}\downarrow 
    && 
    \downarrow^{\mathrlap{\nabla_{B,A}}}
    \\
    C(A\otimes B) &\stackrel{C(\sigma)}{\to}& C(B \otimes A)
  }
\end{displaymath}
commutes, where $\sigma$ denotes the symmetry isomorphism in $sAb$ and $Ch_\bullet^+$.

\end{prop}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item The Eilenberg-Zilber map induces a functor from [[simplicial Lie algebras]] to [[dg-Lie algebras]] (see \href{dgLieAlgebraOfASimplicialLieAlgebra}{here})

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Alexander-Whitney map]]


\item \textbf{Eilenberg-Zilber map}


\item [[Eilenberg-Zilber theorem]]


\item [[cap product]], [[cup product]]



\end{itemize}
In the context of filtered spaces $X_*, Y_*$ and their associated [[FundamentalCrossedComplex|fundamental crossed complex]]es $\Pi X_*, \Pi Y_*$ there is a natural Eilenberg-Zilber morphism

\begin{displaymath}
\eta: \Pi X_* \otimes \Pi Y_* \to \Pi (X_* \otimes Y_*)
\end{displaymath}
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.

\hypertarget{references}{}\subsection*{{References}}\label{references}

The Eilenberg-Zilber map was introduced in (5.3) of

\begin{itemize}%
\item [[Samuel Eilenberg]], [[Saunders MacLane]], On the groups $H(\Pi,n)$, I, Ann. of Math. (2) 58, (1953), 55--106. (\href{https://www.jstor.org/stable/1969820}{jstor})

\end{itemize}
See also

\begin{itemize}%
\item [[Peter May]], 29.7 of \emph{Simplicial objects in algebraic topology} , Chicago Lectures in Mathematics, Chicago, (1967) (\href{http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu}{djvu})


\item [[Andy Tonks|A.P. Tonks]], \emph{On the Eilenberg-Zilber Theorem for crossed complexes}. J. Pure Appl. Algebra, 179{\tt \symbol{126}}(1-2) (2003) 199--220,


\item [[Tim Porter]], section 11.2 of \emph{[[Crossed Menagerie]]},


\item [[Jean-Louis Loday]], section 1.6 of \emph{Cyclic Homology}, Grund. Math. Wiss. 301, Springer, 1992.


\item [[Dan Quillen]], part I, section 4 of \emph{Rational homotopy theory}, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (\href{http://www.jstor.org/stable/1970725}{JSTOR})



\end{itemize}
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

\begin{itemize}%
\item [[Ronnie Brown]], \emph{The twisted Eilenberg-Zilber theorem}. Simposio di Topologia (Messina, 1964), Edizioni Oderisi, Gubbio (1965), 33--37. \href{http://pages.bangor.ac.uk/~mas010/pdffiles/twistedez.pdf}{pdf}

\end{itemize}
[[!redirects shuffle map]]



\end{document}