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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.
Thanks.
This here has a useful bit of discussion of the relation between spectra with G-action and genuine G-spectra in this text here
(mostly around p. 14).
I have started something minimal at spectrum with G-action now, and should add something to G-spectrum now.
I have created also G-spectrum and representation sphere with some minimum content, just for completeness. Cross-linked with relevant entries.
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 $G$-equivariant homotopy theory is an (∞,1)-topos.
By (Rezk1 4) $G Top$ is also the base (∞,1)-topos of the cohesion of the globale equivariant homotopy theory sliced over $\mathbf{B}G$.
The stabilization of the (∞,1)-topos $G Top \simeq PSh_\infty(Orb_G)$ is the equivariant stable homotopy theory of naive G-spectra.
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