Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 12th 2014
   • (edited Apr 12th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAm 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2014
   • (edited Apr 13th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis 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](http://www.maths.ed.ac.uk/~aar/papers/carlsson1.pdf)) (mostly around p. 14). I have started something minimal at _[[spectrum with G-action]]_ now, and should add something to _[[G-spectrum]]_ now.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have created also _[[G-spectrum]]_ and _[[representation sphere]]_ with some minimum content, just for completeness. Cross-linked with relevant entries.

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2014
   • (edited Apr 13th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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: *** ### $(\infty,1)$-topos By [[Elmendorf's theorem]] the $G$-[[equivariant homotopy theory]] is an [[(∞,1)-topos]]. By ([Rezk1 4](#Rezk14)) $G Top$ is also the [[base (∞,1)-topos]] of the [[cohesion]] of the [[globale equivariant homotopy theory]] [[slice (∞,1)-topos|sliced]] over $\mathbf{B}G$. ### Stabilization The [[stabilization]] of the [[(∞,1)-topos]] $G Top \simeq PSh_\infty(Orb_G)$ is the [[equivariant stable homotopy theory]] of [[naive G-spectra]].

   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:

   ---

   $(\infty,1)$-topos

   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 $\mathbf{B}G$.

   Stabilization

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