See the definition of the Alexander-Whitney transformation:


<http://ncatlab.org/nlab/show/Alexander-Whitney+map>


and the Eilenberg-Zilber transformation:


<http://ncatlab.org/nlab/show/Eilenberg-Zilber+map>


Is there a natural or obvious way to extend them to higher tensor powers i.e to, lets say


$$
 \Delta\_{A\_1,\ldots,A\_n} : C(\otimes\_{j=1}^n A\_j) \to \otimes\_{j=1}^n C(A\_j)
$$
and 
$$
 \nabla\_{A\_1,\ldots,A\_n} : \otimes\_{j=1}^n C(A\_j) \to C(\otimes\_{j=1}^nA\_j) 
$$


or to the infinite tensor power series, such that the adjointness is still there?


(My first obvious guess is to simply iterate them using associativity of the usual tensor product but I'm not sure if it is that simple due to concerns about braiding and singns)



Extend Alexander-Whitney and Eilenberg-Zilber map to n-fold tensor products
Yes, there is a generalization to a finite number of simplicial complexes. A reference is *Corollary 2.2* in the paper



> 
> Eilenberg, MacLane: On the groups $H(\Pi,n)$, II: Methods of Computation, Ann. of. Math. 60(1954), No. 1, 49 - 139.
> 
> 
> 


Using the definitions from nLab, the maps are given as follows: 


1) Let $a\_i \in A\_i$ be homogen. 
$$\nabla(a\_1 \otimes \cdots \otimes a\_n) = a\_1 \nabla a\_2 \nabla ... \nabla a\_n$$
(well-definied since $\nabla$ is associative)


2) Let $a\_i \in (A\_i)\_m$. 
$$\Delta(a\_1 \otimes \cdots \otimes a\_n) = \sum \displaystyle \otimes\_{i=1}^n\displaystyle\tilde{d}^{m-j\_i}d\_0^{j\_{i-1}}a\_i$$ 
where the sum is taken over $0 \le j\_1 \le \cdots \le j\_{n-1} \le m$ and $\tilde{d}^{m-j\_n},\;d\_0^{j\_0}$ has to be interpreted as identity. 


