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I'm looking for a comprehensive reference (for citation purposes) laying out the basic facts of the equivalence between $G$-spaces and bundles over $BG$ for a discrete group $G$. I'd like it to also include the identifications of the various auxiliary structures: $G$-representations to local systems, group cohomology to cohomology with local coefficients, (Borel) equivariant cohomology and cohomology of the total space of the bundle, the Hochschild-Serre spectral sequence and the Serre spectral sequence of the fibration.

This seems like foundational material but I've found this shockingly hard to find in a reference; can someone help me out?

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One reference is the two papers by Nikolaus–Schreiber–Stevenson:

Principal ∞-bundles – General theory

Principal ∞-bundles – Presentations

In particular, these papers explain the equivalence between G-spaces, spaces over BG, and principal G-bundles. They also cover twisted cohomology, twisted bundles, gerbes, etc.

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    $\begingroup$ These papers seem like they may be more infinity-y than the original asker may have had in mind. I'd also be interested in a more classical reference where spaces are just spaces. $\endgroup$
    – Greg Friedman
    Commented Jul 6, 2023 at 1:11
  • $\begingroup$ @GregFriedman: The second paper makes no use of ∞-categories and is formulated in the traditional language. $\endgroup$
    – Dmitri Pavlov
    Commented Jul 6, 2023 at 5:04
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    $\begingroup$ Hmm, I don't mean to be disagreeable, but I just looked at the second paper, and it also seems to be largely about presentations of ∞-toposes using things like simplicial presheaves, simplicial objects in sites, etc. The section labeled "groups" starts off with "Every ∞-topos H comes with a notion of ∞-group object that generalizes the ordinary notion of group object in a topos as well as that of grouplike A∞ space in Top ≃ Grpd∞. We discuss presentations of ∞-group objects by presheaves of simplicial groups." So, we may have very different interpretations of "traditional"! $\endgroup$
    – Greg Friedman
    Commented Jul 7, 2023 at 6:42
  • $\begingroup$ @GregFriedman: Simplicial groups were introduced in 1950s (Kan) and simplicial presheaves in 1980s (Joyal, Jardine), seems reasonably traditional for a topic in homotopy theory. The OP requested a source for reference purposes, which these papers can certainly provide. $\endgroup$
    – Dmitri Pavlov
    Commented Jul 7, 2023 at 7:27

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