Associated to a DGCA (differential graded commutative algebra) $A$, we can associate to it the category of modules over $A$. Hence, for a space $X$ we can consider the category of modules over $\Omega\_{PL}(X)$, modules over the PL de Rham complex of $X$. The starting point of rational homotopy theory is that $\Omega\_{PL}(-)$ is an equivalence upon restricting to simply connected spaces and simply connected DGCA's in a homotopical sense.


I am wondering, perhaps after replacing $\Omega\_{PL}(X)$ with a Sullivan model, does this category of modules have a topological interpretation? I am also interested in the d.g. lie algebra analog.



Is there a topological interpretation of a module over $\Omega_{PL}(X)$?
TLDR: there is a contravariant adjunction between the derived category of $A\_{PL}(BG)$ and the naive category of rational $G$-spectra. In some cases, this is a contravariant equivalence of categories, for example when $G$ is a connected Lie group.




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If I am not mistaken, the derived category of $A\_{PL}(X)$ is equivalent to the homotopy category of modules over the ring spectrum $\operatorname{Map}(\Sigma^\infty X\_+, H\mathbb Q)$. I think this follows from a paper of Richter and Shipley.


We may as well pose the question about modules over the singular cochain complex $C^\*(X)$, a.k.a modules over $\operatorname{Map}(\Sigma^\infty X\_+, H\mathbb Z)$, or even more generally, about modules over the ring spectrum $D(X)=F(\Sigma^\infty X\_+, S)$ (the Spanier-Whitehead dual of $X\_+$).


When $X$ is a disjoint union, all these rings or ring spectra split as products. So we may assume that $X$ is connected. In this case there is a Koszul duality between cochains on $X$ and chains on $\Omega X$. Or better still, there is a Koszul duality between the ring spectra $D(X)$ and $\Sigma^\infty \Omega X\_+$. This implies that there is a contravariant adjunction between the homotopy categories of modules of these spectra. This adjunction restricts to an equivalence between certain subcategories of these module categories. For example, there is an equivalence between finitely generated free cellular modules over either one of the ring spectra, and the so-called nilpotent modules over the other one.


Without loss of generality we may assume that $X=BG$ where $G$ is a topological groups. Then modules over $\Sigma^\infty \Omega BG\_+\simeq \Sigma^\infty G\_+$ are the same as spectra with an action of $G$. So there is a contravariant adjunction between the category of modules over $D(BG)$ and the naive category of $G$-spectra. Similarly, there is a Koszul duality between the ring spectra $D\_{\mathbb Q}(BG)=\operatorname{Map}(\Sigma^\infty BG\_+, H\mathbb Q)$ and $H\mathbb Q \wedge G\_+$. If I am not mistaken, the homotopy category of modules over $\Omega\_{PL}(X)$ is equivalent to modules over $D\_{\mathbb Q}(BG)$. So we obtain a contravariant adjunction between this category and the naive category rational $G$-spectra.


This adjunction is not an equivalence in general, but sometimes it is. For example, I think the paper


"[An algebraic model for free rational $G$-spectra for connected compact Lie groups](https://arxiv.org/abs/0906.5575)", by Greenlees and Shipley,


tells you that it is an equivalence when $G$ is a connected compact Lie group. They also give an explicit algebraic model for the category of modules in this case.


I have the feeling that when $G$ is connected and $BG$ is finite, the adjunction is also close to being an equivalence in some sense that can be made precise. Maybe the (naive) category of rational $G$-spectra is equivalent in this case to the category of pro-nilpotent modules over $A\_{PL}(BG)$, but I am not 100% sure. There is a paper about derived Koszul duality by Blumberg and Mandell that says something about it.


On the other hand, if $G$ is a finite group, then $D\_{\mathbb Q}(BG)\simeq H\mathbb Q$, but the category of rational $G$-spectra of course depends on $G$.


(Incidentally the paper


[Complexes of injective $kG$-modules](https://projecteuclid.org/journals/algebra-and-number-theory/volume-2/issue-1/Complexes-of-injective-kG-modules/10.2140/ant.2008.2.1.full) by Benson and Krause contains some interesting information about the derived category of $C^\*(BG; \mathbb F\_p)$ for finite $G$.)


