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I should say – for those watching the logs and wondering – that I started editing the entry global equivariant homotopy theory such as to reflect Charles Rezk’s account in a coherent way.
But I am not done yet. The entry has now some of the key basics, but is still missing the general statement in its relation to orbispaces. Also some harmonizing of the whole entry may be necessary now, as I moved around some stuff.
So better don’t look at it yet. I hope to bring it into shape tomorrow or so.
(In the process I have split off global orbit category now.)
As far as I know, Elmendorff’s Theorem applies to any topological group so I don’t see what’s the advantage of restricting to Lie groups. But more importantly, I have to say that I was wondering for a long time why Top cannot be replaced with sSet. If we replace a topological group by a simplicial group, is there a family of sub-simplicial groups that is suitable to replace the family of closed subgroups?
Another remark is that locally, and hence globally, there are two kinds of equivariant homotopy theory, usually called strong and weak, or Bredon and Borel. A model structure for global Borel equivariant homotopy theory is spelled out in “The Grothendieck construction for model categories”.
Hi Matan,
regarding Lie groups: I need the assumption of Lie groups for some of the statements on that page, but not for all. I should go through the entry and sort this out more.
regarding simplicial groups: so for equivariant homotopy theory it is crucial that $G$ is really regarded as a topological group (a geometric group object) and not as its underlying groupal homotopy type. That’s why the theory knows both about the genuine fixed points as well as the homotopy quotient (whereas only the latter would be defined for groupal homotopy types). This “genuinely geometric” aspect of equivariant homotopy theory is formally precisely what is captured by Charles Rezk’s observation that the global equivariant homotopy theory is cohesive.
regarding Borel- and Bredon- equivariance: this is discussed at equivariant cohomology. But I’ll add more cross-links to that now to highlight it more.
Thanks Urs! I still wonder about a naive question: are you claiming that if I take all topological groups instead of just the compact Lie ones, then the global (strong) equivariant homotopy theory would not be a coheseve topos?
At least the proof as currently written out (here) by Charles Rezk does use the assumption. Most explicitly so in the proof of lemma 6.2.2 (p. 20). (This lemma is however needed “only” for the product-preservation of the extra left adjoint.)
I have added a pointer to your article with Yonatan to global equivariant homotopy theory, also pointing to Borel model structure.
Thanks!
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