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For purposes of $G$-equivariant rational homotopy theory one wants a Quillen adjunction which generalizes the classical one of Bousfield-Gugenheim from plain dg-algebras/simplicial-sets to (co-)presheaves of such over the $G$-orbit category. This has been produced in:

where the expected RHT equivalence on homotopy categories is proven for equivariantly simply-connected spaces/algebras.

To get around restrictions on equivariant $\pi_1$, I am wondering whether this model structure might transfer from the plain orbit category to tom Dieck's "fundamental category" $\Pi_G(X)$ of a given $G$-space $X$: This is the category fibered over the $G$-orbit category, $\Pi_G(X) \xrightarrow{p} G \mathrm{Orbt}^{\mathrm{op}}$, with fiber over $G/H$ being the fundamental groupoid of the fixed subspace $X^H$.

Here I mean "transfer" both colloquially as well as technically: Generally I'd be interested in a model structure on dg-algebra valued functors on $\Pi_G(X)$ which is analogous to Scull's model structure for such functors on $G \mathrm{Orbt}$. But more concretely I am wondering if the latter left-transfers, in the technical sense, along $p_!$.

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I am aware of the usual sufficient conditions to be checked, but it seems fiddly and I feel like lacking a good strategy to approach this. Some vague thoughts:

I notice that Scull shows (around Cor. 3.12) much, though not all, of what it would take for her model structure to be cofibrantly generated, which might help.

Also, it is probably noteworthy that $\Pi_G(X)$ is still an EI-category, which seems to be the main property of $G\mathrm{Orbt}$ that Scull's proofs rely on (incidentally, is there a more transparent proof of her Prop. 2.5, quoted from Prop. 3.4 in Triantafillou 1982?). This is reminiscent of Golasiński 1997, who discusses minimal systems of dg-algebra models in the generality over EI-categories.

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  • $\begingroup$ It seems like it would be a good idea to pin down the details of the case $G=1$ first. Even there you dont expect to get a Quillen equivalence without some nilpotency condition on $\pi_1$. $\endgroup$
    – Mark Grant
    Commented Jul 21, 2022 at 9:15
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    $\begingroup$ @MarkGrant thanks for commenting. Let me highlight that the transfer is not meant to be a Quillen equivalence. For $G = 1$ it's meant to give the $\pi_1(X)$-equivariant Borel model structure. For general $G$ it must be a mix between proper $G$-equivariance and $\pi^{H}_1(X)$-Borel equivariance, as $H \subset G$ varies. (Incidentally, I can see that the transfer model structure exists under the assumption that $G/H \mapsto \pi^{H}_1(X)$ is projectively cofibrant. Unfortunately, that's too strong a condition to be of much interest.) $\endgroup$
    – Urs Schreiber
    Commented Jul 21, 2022 at 19:47

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