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Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric) realization of a simplicial group (re-topologize this using the compactly generated topology).

Let $R^G(\ast)$ be the category of based left $G$-spaces (where spaces are taken in $T$). Then $R^G(\ast)$ is a model category in which a fibration and weak equivalence are defined via the forgetful functor to $T$. Cofibrations are defined by the lifting property. It's well-known that this gives a model structure, so I'll take that for granted.

Let $R(BG)$ be the category of spaces containing $BG$ as a retract. Objects are spaces $Y$ equipped with maps $r:Y \to BG$, $s: BG \to Y$ such that $r\circ s : BG \to BG$ is the identity (call $r$ and $s$ structure maps). A morphism $Y \to Y'$ is a map of underlying spaces that preserves the structure maps.

Then $R(BG)$ is a model category in which a fibration, cofibration and weak equivalence are defined using the forgetful functor to $T$. This is due to Quillen.

I think the following is a folklore result:

Assertion: $R^G(\ast)$ and $R(BG)$ are Quillen equivalent.

My question: Does anyone know a concrete reference for this?

Remark: A statement suggesting that the assertion is true in the context of Waldhausen categories appears in Waldhausen's foundational paper in LNM 1126.

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Appendix A in this paper seems to do this, unless I've misunderstood:

http://arxiv.org/abs/0810.4535

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    $\begingroup$ It seems that article deals with unbased $G$-simplicial sets. $\endgroup$
    – Ricardo Andrade
    Commented May 21, 2013 at 23:43
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    $\begingroup$ On the other hand, the above article makes a reference to "Parametrized spaces model locally constant homotopy sheaves" (arxiv.org/abs/0706.2874) by Michael Shulman. Corollary 8.7 in this article is very close to the result that John Klein is asking for. $\endgroup$
    – Ricardo Andrade
    Commented May 22, 2013 at 0:04
  • $\begingroup$ Fair, though I would be surprised if the proof of the based case was much different. $\endgroup$
    – Dylan Wilson
    Commented May 22, 2013 at 0:56
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    $\begingroup$ @Ricardo: Thanks. I think you're right---the Shulman result is the one I want to reference. Since that Dylan's answer prompted your comment, I guess I should give Dylan the credit for the answer. $\endgroup$
    – John Klein
    Commented May 22, 2013 at 2:01
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    $\begingroup$ @Dylan: That is true. If I read the relevant passage in Michael Shulman's article correctly, he derives the based case from the unbased one using a nice, fairly elementary model categorical lemma found in Hovey's book. In any case, the result for spaces (as opposed to simplicial sets) really should be attributed to Shulman. Thank you for bringing up the article "Units of ring spectra and Thom spectra" making use of Shulman's result. // @John: You are welcome. Also, I would absolutely not have found this article without Dylan's answer, so the credit is his. $\endgroup$
    – Ricardo Andrade
    Commented May 22, 2013 at 4:33

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