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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.

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 rings, and the so-called nilpotent modules over the other ring.

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 (contrary to what I wrote initially), but sometimes it is. For example, I think the paper

"An algebraic model for free rational $G$-spectra for connected compact Lie groups", 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.

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 by Benson and Krause contains some interesting information about the derived category of $C^*(BG; \mathbb F_p)$ for finite $G$.)

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.

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", 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 by Benson and Krause contains some interesting information about the derived category of $C^*(BG; \mathbb F_p)$ for finite $G$.)

This is a much revised version of my initial answer, which had some blunders. The TLDR version: 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.

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 rings, and the so-called nilpotent modules over the other ring.

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 (contrary to what I wrote initially), but sometimes it is. For example, I think the paper

"An algebraic model for free rational $G$-spectra for connected compact Lie groups", 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.

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 $G$.

Then again, I think that results of Blumberg and Mandell in their paper on derived Koszul duality tell you that the adjunction is an equivalence if $G$ is connected and $BG$ is a finite complex.

To summarize, I suspect that the derived category of $A_{PL}(BG)$ is equivalent to the naive category of rational $G$-spectra if $G$ is connected, and either $G$ or $BG$ is a finite complex.

(Incidentally the paper

Complexes of injective $kG$-modules by Benson and Krause contains some interesting information about the derived category of $C^*(BG; \mathbb F_p)$ for finite $G$.)

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.

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 rings, and the so-called nilpotent modules over the other ring.

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 (contrary to what I wrote initially), but sometimes it is. For example, I think the paper

"An algebraic model for free rational $G$-spectra for connected compact Lie groups", 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.

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 by Benson and Krause contains some interesting information about the derived category of $C^*(BG; \mathbb F_p)$ for finite $G$.)

This is a much revised version of my initial answer, which had some blunders. The TLDR version: 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.

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 rings, and the so-called nilpotent modules over the other ring.

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 (contrary to what I wrote initially), but sometimes it is. For example, I think the paper

"An algebraic model for free rational $G$-spectra for connected compact Lie groups", 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.

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 by Benson and Krause contains some interesting information about the derived category of $C^*(BG; \mathbb F_p)$ for finite $G$.)

This is a much revised version of my initial answer, which had some blunders. The TLDR version: 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.

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 rings, and the so-called nilpotent modules over the other ring.

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 (contrary to what I wrote initially), but sometimes it is. For example, I think the paper

"An algebraic model for free rational $G$-spectra for connected compact Lie groups", 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.

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 $G$.

Then again, I think that results of Blumberg and Mandell in their paper on derived Koszul duality tell you that the adjunction is an equivalence if $G$ is connected and $BG$ is a finite complex.

To summarize, I suspect that the derived category of $A_{PL}(BG)$ is equivalent to the naive category of rational $G$-spectra if $G$ is connected, and either $G$ or $BG$ is a finite complex.

(Incidentally the paper

Complexes of injective $kG$-modules by Benson and Krause contains some interesting information about the derived category of $C^*(BG; \mathbb F_p)$ for finite $G$.)