The stable Dold-Kan correspondence says that for every commutative ring $R$, there is an equivalence of $\infty$-categories between the category $Ch(R)$ of (unbounded) chain complexes of $R$-modules and the category $Mod(HR)$ of module spectra over the Eilenberg-Maclane ring spectrum $HR$. This can be realized as a Quillen equivalence of suitable model categories (see [here](http://ncatlab.org/nlab/show/stable+Dold-Kan+correspondence)).


My question is whether there is a *parameterized* version of this equivalence. Namely, given a based space $B$, there is a way to construct the category of parametrized spectra over $B$ with a suitable model structure (see [here](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf)). In this setting there is a monoidal structure given by smash product "over $B$". We therefore can construct the parametrized ring spectrum $HR\times B$ and I guess also the category of parametrized modules over it (with suitable model structure). I wonder if this is equivalent (as an $\infty$-category) to something we can construct on the level of chains. 


My guess is that it should be equivalent to something like co-modules over the co-algebra $C\_\* (B)$ in $Ch(R)$. Is there indeed some algebraic model like this?



Parametrized Dold-Kan correspondence?
We Discussed this privately but for future generations let me write this here.
This is false. As an example take the $B$ to be the classifying space of an acyclic group. Then $C\_\*(B) = \mathbb{Z}$. So if what you write was true, every local system on such a space would have been constant. but this is false. For example consider the regular representation.


