# Parametrized Dold-Kan correspondence?

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).

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). 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?

• Your parametrized chain complexes and ring spectra are just categories of $(\infty,1)$-functors from $B$ to $Ch(R)$ or $HR-Mod$. So of course these two categories are also equivalent. Oct 22, 2015 at 16:14
• Actually, the exact statement depends on how you consider $B$. I was assuming you mean $B$ as a homotopy type. Then on one side there is an $\infty$-category of locally trivial $\infty$-sheaves of $R$-module chain complexes and on the other side $HR\times B$ modules in fibrations of spectra over $B$. Similarly if you don't want locally trivial fibrations, but general sheaves. Oct 22, 2015 at 16:20
• Thank you Anton. I understand your perspective, but it's not clear to me how to actually show that sheaves of chain complexes of $R$-modules on $B$ are the same as co-modules over the co-algebra $C_*(B)$. If you could perhaps expand on this in an answer I would gladly accept it. Oct 23, 2015 at 10:52
• @Kotel: if you're looking for something really "algebraic" then the thing you want is module spectra over the "group algebra" $R[\Omega B]$ (if $B$ is connected). A relationship to $C_{\bullet}(B, R)$ would be a form of Koszul duality and should require some finiteness hypotheses, and it would probably help if $B$ were simply connected as well. Oct 26, 2015 at 7:04

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.