Timeline for Homotopy pullback of Quillen equivalence
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| Apr 30, 2018 at 9:38 | comment | added | David White | @LennartMeier, thanks for your insightful response. I proceeded along the same lines, but got stuck at the part you said "should be classical." Anyway, I learned today that, even if this is true, it doesn't help, so I'm going to stop thinking about it. I still believe this MO question is interesting, and I hope it gets a full answer at some point. As for Bergner and localization, Thm 3.1 in her "homotopy fiber products" paper seems best-possible to me. Combinatorial is probably necessary. | |
| Apr 27, 2018 at 9:07 | comment | added | Lennart Meier | ...pushouts and pullbacks. Bergner and the more general staff.science.uu.nl/~meier007/HomotopyColimits2.pdf thus show these constructions to be homotopy pushouts/pullbacks in relative categories. Btw: Bergner does not really give a general construction of the localized model structure on $M$; do you have one? | |
| Apr 27, 2018 at 8:58 | comment | added | Lennart Meier | If you have a homotopy pushout/pullback along a weak equivalence in relative categories, it is a weak equivalence again. Quillen equivalences should define weak equivalences of the underlying relative categories; by Barwick-Kan 'A characterization of simplicial localization functors and a discussion of DK equivalences' we just have to show it induces an equivalences on hammock localizations (i.e. on mapping spaces), which should be classical. As the Rezk classifying diagram from relative categories to complete Segal spaces is an equivalence of homotopy theories, it detects homotopy.... | |
| Apr 26, 2018 at 21:35 | history | asked | David White | CC BY-SA 3.0 |