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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 7th 2013
• (edited Oct 7th 2013)

started Elmendorf’s theorem with a brief statement of the theorem

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 15th 2014

added a section Model category presentation / Quillen equivalence with some brief paragraphs on Guillou’s notes (fro discrete groups).

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeJun 28th 2018

I added a remark about the generalisation to atomic orbital $(\infty, 1)$-categories of Barwick et al.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJun 28th 2018

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.

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeJun 29th 2018
• (edited Jun 29th 2018)

Added a reference to the n-cat café discussion and also to an old paper on the categorical aspects of the theorem by Cordier and myself. (PS. This includes an attempt at saying why, intuitively, the theorem works by looking at G-sets and diagrams of sets over Orb(G).)

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeJun 29th 2018

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeJul 3rd 2018

Added link to Marc Stephan’s note proving a Quillen equivalence for arbitrary families of subgroups.

• CommentRowNumber8.
• CommentAuthorTim_Porter
• CommentTimeJul 4th 2018

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?

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeJul 4th 2018

I’ve shifted the generalizations mentioned in two remarks to the section ’Generalizations’.

What is now generalization 3 is referring to the work of Farjoun and others that Tim has just mentioned.

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeJul 4th 2018

Apparently the latest on this is:

• B. Chorny, Homotopy theory of relative simplicial presheaves, Israel J. Math. 205(2015), no. 1, 471–484.
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJul 4th 2018
• (edited Jul 4th 2018)

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.

• CommentRowNumber12.
• CommentAuthorTim_Porter
• CommentTimeJul 4th 2018

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!

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeMay 16th 2019