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One thing I am thinking that we know we have a map $Sp \to Ho(Sp)$. Since $Ho(Sp)$ is an additive category so we can add maps. Therefore, we can take the alternative sum of face maps in $Ho(Sp)$ then take a representative in $Sp$ and define that as the differential. Now if we have a cosimplicial object $W^\bullet$ in $Sp$. Here each $W^n$ is an $E$-module spectra. Then we have a map $W^n \to M^n W^\bullet$, the Matching object. Then can we claim that the kernel of the map $W^n \to M^n W^\bullet$ is a summand of $W^n?$ I think the kernel should be $\bigcap_{i=0}^{n-1} ker(s^i).$
Btw, I don't need a formal Dold-Kan equivalence: all I need is a procedure for producing a chain complex of spectra from a simplicial object in symmetric spectra.
For any pointed simplicial category we can define the chain compexes are those such that composition of consecutive differentials is nullhomotopic. We can do this for the category of symmetric spectra. With these notions in our hand can we say something about the normalized complex functor or the Moore complex functor from simplicial objects in symmetric spectra to the chain complexes in symmetric spectra?
@DylanWilson: Can you please give me a reference for " There is an equivalence between the homotopy theory of filtered objects and simplicial objects in spectra"?
@DavidWhite: My precise question is: Is there any categorical equivalence or Quillen equivalence between the category of simplicial objects in symmetric spectra, $s(Sp^\Sigma)$, and category of chain complexes in $Sp^\Sigma$?
@Prasit: Thank you so much. Yeah, I know if a spectrum is smashing then that definitely sits inside the list. Using 10.6 and 10.9 of Ravenel's paper on "Localization of spectra with respect to certain periodic homology" it follows that E(n) is smashing. But do you know more examples?
There is a natural map $f : S(\xi)_+ \wedge S^V \rightarrow S(\xi^j)_+ \wedge S^V$. Then for what $n$, $\tilde{H}^n_{Z/n}(f)$ is nonzero?Where $V$ is a representation of $Z/n$ and $f(z,v) = (z^j ,v) , z \in S(\xi) , v \in S^V$.
:What can you say about the convergence when $X$-is $S(\xi)$? Where $\xi$ is the irreducible representation of $Z/n$ given by multiplication by $e^{2i \pi /n}$
@SteveCostenoble : Thank you for your valuable comments. But it'll be great if you can produce some formula of Kunneth type for constant coefficient sysmtem.
