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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?

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    $\begingroup$ 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. $\endgroup$ – Anton Fetisov Oct 22 '15 at 16:14
  • $\begingroup$ 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. $\endgroup$ – Anton Fetisov Oct 22 '15 at 16:20
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    $\begingroup$ 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. $\endgroup$ – KotelKanim Oct 23 '15 at 10:52
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    $\begingroup$ @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. $\endgroup$ – Qiaochu Yuan Oct 26 '15 at 7:04
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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.

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    $\begingroup$ Maybe we should re-ask the question restricted to simply-connected spaces. If we take chains with rational coefficients, then that's the standard setup of rational homotopy theory, and the question becomes whether rationalized parameterized spectra over simply connected spaces (maybe we also want finite type) are equivalent to dg-modules over the Sullivan algebras of the base spaces. That question I just asked here mathoverflow.net/q/261747/381 (and only now did I find your discussion here). $\endgroup$ – Urs Schreiber Feb 9 '17 at 11:30

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