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started Elmendorf’s theorem with a brief statement of the theorem
added a section Model category presentation / Quillen equivalence with some brief paragraphs on Guillou’s notes (fro discrete groups).
This was sparked by John Huerta’s recent Café post where he’s seeking a “nice conceptual explanation for Elmendorf’s theorem”. You’d imagine that a very general setting would provide this.
But now I look again at that introductory paper at Parametrized Higher Category Theory and Higher Algebra, what precisely are they promising in an Elmendorf direction? On p. 6 certain unstable results are said to hold for any base $(\infty, 1)$-category, while stable results need those ’atomic’ and ’orbital’ properties.
An “Elmendorf–McClure theorem” is discussed on p. 8.
Added doi link, tweaked formatting of (Cordier-Porter 1996)
There are some papers by Dror that are sometimes cited.
E. Dror Farjoun, Homotopy theories for diagrams of spaces, Proceedings of the American Mathematical Society, 101, (1987), 181 – 189.
E. Dror Farjoun and A. Zabrodsky, Homotopy equivalence between diagrams of spaces, Journal of Pure and Applied Algebra, 41, (1986), 169 – 182.
These could be mentioned in several places (but do not seem to be there, of course, I may have searched on the worng term!) ELmendorf’s theorem, Orbit category etc. They are cited by Barwick et al. and are relevent here as well. They could be useful but where?
Apparently the latest on this is:
I have grouped the reference Stephan 10 together with Stephan 13 and expanded the citation data (MS thesis 2010, and pointer also to the full thesis text). I checked briefly if the latter is just the published version of the former, but maybe not.
I seem to recall that the generalization to sub-families of subgroups is also claimed in May 96, just not in terms of model categories. So I have added a “see also” to May96. If anyone has the energy to check, we should add pointer to the precise proposition number.
I have a vague memory the Dwyer and Kan looked at the sub-families of subgroups quite early as well… but they did such a lot in the 1980s that I am not sure where to look!
I notice that our entry never says why one would want the equivariance group $G$ to be a compact Lie group.
Curiously, neither does Elmendorf in his article! He demands this condition up front, but never ever refers back to it, explicitly.
Of course it’s a sufficient condition to ensure that with a $G$-space $X$ being an equivariant CW-complex, so are all its fixed loci $X^H$ for closed subgroups $H$. This is – implicitly – used deep down in the proof of the equivalence.
Marc Stephan 10 claims that this is shown to work without any condition on $G$ in May 96. But in May 96 it’s not easy to see where the exact assumptions are stated, and when it comes to using them (p. 53) it just says “We can check”.
I have adjusted the statement in the Idea-section back to finite groups, and then added the following remark:
This is stated in Elmendorf 83 as an equivalence of homotopy categories and refined in Guillou 06, Prop. 3.15 to a Quillen equivalence of presenting model categories.
Notice that Elmendorf 83 allows $G$ more generally to be a compact Lie group, while Piacenza 91, Sec 6 and May 96, Sec. V.3 claim that the equivalence of homotopy categories works even for any topological group.
However, Guillou 06 assumes $G$ to be a finite group to get an actual Quillen equivalence (Guillou 06, Prop. 3.15).
While Stephan 13 claims that Piacenza 91 also gives a Quillen equivalence, this is not what Piacenza 91, Thm. 6.3 actually states. (What is stated certainly goes in the direction of claiming that the derived adjunction is an equivalence, but does it go all the way?)
added pointer to:
This has the infinity/Quillen equivalence version but again stated just for discrete groups, (albeit not assumed finite).
added also pointer to
Then I finally saw that somebody had added a pointer to John Huerta talking about Elmendorf’s theorem. I found that odd without the pointer to where this came from, so I added
Then I spotted pointer to
with the mysterious remark that
Some of the categorical aspects of Elmendorf’s theorem are examined…
Looking into the article, I see that Theorem 3.11 there enhances Elmendorf’s equivalence to a simplicial adjunction that is at least very close to a simplicial Quillen equivalence. Therefore I moved this reference up to the list of references on $\infty$-enhancements of Elmendorf, gave it a (hopefully) more informative commentary and linked to it from the main text accordingly.
Then I finally looked at the reference to Chorny’s article that was (and is) given in the entry. Noticing that this is really a note expanding on previous articles, I added those precursors and with more (hopefully) informative commentary. Now it reads as follows:
A more general class of Quillen equivalences of which these model-category theoretic enhancements of Elmendorf’s theorem turn out to be special cases are discussed in:
William Dwyer, Daniel Kan, Section 2 of: Singular functors and realization functors, Indagationes Mathematicae (Proceedings) Volume 87, Issue 2, 1984, Pages 147-153 (doi:10.1016/1385-7258(84)90016-7)
Emmanuel Dror Farjoun, Prop. 1.3 in: Homotopy Theories for Diagrams of Spaces, Proceedings of the AMS, Vol. 101, No. 1 (Sep., 1987), pp. 181-189 (jstor:2046572 )
{#Chorny13} Boris Chorny, Homotopy theory of relative simplicial presheaves, Israel J. Math. 205 (2015), no. 1, 471–484, (arXiv:1310.2932)
Hm, that made me realize that the statement in its full beauty is already way back in
They already have a full-blown simplicial Quillen adjunction and no assumption on the topological group $G$.
Have re-written the commentary on the literature accordingly.
I am working on a remark (here) which means to bring out more of the generality of the theorem that Dwyer & Kan actually proved (which goes far beyond what is traditionally cited) and what that really “means” (to my mind).
I don’t claim this remark is well-written yet, please to be regarded as under construction. But there are two points that deserve to be made:
Dwyer-Kan prove a result that holds for every topological group $G$ and every choice of familiy $\mathcal{F}$ of subgroups.
If this choice is different from $G = CompactLieGrps$ and $\mathcal{F} = ClosedSubgroups$ then the resulting homotopy theory is not (or not known/guaranteed to be) that of $G$-spaces with usual $G$-homotopies, but
since their result is a simplicial Quillen equivalence, there is always a statement just about mapping spaces, and that statement is of the curious form “concordance becomes homotopy” which has been so important elsewhere;
namely, if we understand “$(G,\mathcal{F})$” as always referring to the coset spaces $G/H$ in the given family, then their theorem implies that “shape may be taken inside the mapping space” in the following way:
$X \,\in\, (G,\mathcal{F})CWCplx \;\;\;\;\;\; \vdash \;\;\;\;\;\; ʃ \Big( Maps \big( \mathrm{X} ,\, \mathrm{Y} \big)^G \Big) \;\; \simeq \;\; Psh_\infty\big(Orb(G,\mathcal{F})\big) \Big( ʃ \big(\mathrm{X}^{(-)}\big) ,\, ʃ \big(\mathrm{Y}^{(-)}\big) \Big) \,.$It looks like we have a neat proof of a twisted Elmendorf theorem (I’ll show the proof tomorrow):
For
$G$ a finite equivariance group,
$\Gamma$ a $G$-equivariant topological structure group,
satisfying Uribe & Lück’s “Condition H”,
$X$ a $G$-CW complex (such as a smooth $G$-manifold),
$A$ any $G$-space with $\Gamma$-action,
then for any $G$-equivariant $A$-fiber bundle with structure group $\Gamma$ over $X$,
the concordance $\infty$-groupoid of its $G$-equivariant sections is equivalently
the slice hom of $\infty$-presheaves over $G$-orbits,
from the shape of $X^{(-)}$ to the shape of the $G$-orbi-singularization of $A \sslash (\Gamma \rtimes G)$,
sliced over the equivariant $\Gamma$-classifying space.
$\,$
So for $\Gamma = 1$ this reduces to the usual Elmendorf theorem (in the $\infty$-form that Dwyer & Kan proved),
while for non-trivial $\Gamma$ this is a variant version where both sides are sliced over the appropriate incarnation of $\mathbf{B}(\Gamma \rtimes G)$.
$\,$
Has any such sliced/parameterized/twisted Elmendorf theorem been considered/proven elsewhere?
(I haven’t seen anything remote elsewhere – but if anyone has, please drop a note!)
A first writeup of that proof is now in Section 4.5 of the pdf here.
(Currently it’s Thm. 4.5.3 on p. 223, but this may change.)
The main technical step used in the proof is the observation that under that “Condition H”, the full sub-$\infty$-category of the finite-isotropy orbits of $\Gamma \rtimes G$ (for $\Gamma$ the topological structure group and $G$ the discrete equivariance group) on those which are lifts of subgroups of $G$ is equivalent to the slice of $G$-orbits over the equivariant classifying space:
$Orb(G)_{/B_G\Gamma} \xhookrightarrow{\phantom{---}} Orb(\Gamma \rtimes G) \,.$This is currently Prop. 4.3.15 on p. 211 (but this may change).
This new material still has some rough edges, but should be readable.
Added this pointer:
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