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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 12th 2014
    • (edited Apr 12th 2014)

    Am on my phone and about to be forced offline, and so have trouble searching the literature to remind me: maybe somebody here could help:

    What’s again the precise relation between spectral presheaves on the orbit category and the more refined definitions of G-equivariant spectra, e.g. Mackey functors, etc. There are evident inclusions/restrictions, but what’s their propeties (faithfuness, essential image etc.)?

    In Guillou-May arXiv:1110.3571 the restriction from G-spectra to spectral presheaves on Orb_G is mentioned on p. 6, but again I am looking for intrinsic characterization of what that restriction forgets.


    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2014
    • (edited Apr 13th 2014)

    This here has a useful bit of discussion of the relation between spectra with G-action and genuine G-spectra in this text here

    • Gunnar Carlsson, A survey of equivariant stable homotopy theory,Topology, Vol 31, No. 1, pp. 1-27, 1992 (pdf)

    (mostly around p. 14).

    I have started something minimal at spectrum with G-action now, and should add something to G-spectrum now.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2014

    I have created also G-spectrum and representation sphere with some minimum content, just for completeness. Cross-linked with relevant entries.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2014
    • (edited Apr 13th 2014)

    I keep making small additions, mostly cross-references, to our equivariant entries, to make them eventually give a more coherent story (currently they are still a bit of a mess…):

    have added to Elmendorf’s theorem the remark that it in particular implies that the equivariant homotopy theory is an \infty-topos

    Similarly I have added to equivariant homotopy theory a Properties-section with brief remarks:


    By Elmendorf’s theorem the GG-equivariant homotopy theory is an (∞,1)-topos.

    By (Rezk1 4) GTopG Top is also the base (∞,1)-topos of the cohesion of the globale equivariant homotopy theory sliced over BG\mathbf{B}G.


    The stabilization of the (∞,1)-topos GTopPSh (Orb G)G Top \simeq PSh_\infty(Orb_G) is the equivariant stable homotopy theory of naive G-spectra.