Timeline for Finite homotopy limits commute with sequential homotopy colimits
Current License: CC BY-SA 3.0
8 events
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| Sep 30, 2013 at 18:46 | comment | added | Boris Chorny | I think that you are right. In a more general situation it is probably possible to find a counterexample, but every combinatorial model category is Quillen equivalent, by a theorem of Dugger, to a left Bousfield localization of simplicial presheaves over some small category equipped with the projective model structure. But in such categories generating cofibrations are the same as the projective generating cofibrations. In particular, they have $\aleph_o$-presentable domains and codomains. | |
| Sep 30, 2013 at 15:25 | comment | added | Emanuele Dotto | Boris, thanks for your answer. If $\lambda$ is bigger than $\aleph_0$, is it then not true in general that sequential homotopy colimits commute with finite homotopy limits? (thanks for clarifying what finite means in this context) | |
| Sep 30, 2013 at 1:18 | vote | accept | Emanuele Dotto | ||
| Sep 29, 2013 at 19:05 | comment | added | Boris Chorny | No, this assumption can also be removed using a framing and applying the respective formulas from Hirschhorn's book for the computation of homotopy limits. Alternatively we can replace our combinatorial model category by a simplicial one in a homotopy meaningful way and conclude that finite homotopy limits commute with filtered homotopy colimits since they commute in the simplicial replacement. | |
| Sep 29, 2013 at 13:07 | comment | added | Fernando Muro | Sorry, Boris, I meant simplicial, not cofibrant. I don't know why I wrote cofibrant. | |
| Sep 29, 2013 at 11:50 | comment | added | Boris Chorny | Of cause not, Fernando, cofibrant generation is a luxury. We just need it to compare the homotopy filtered colimits with the strict filtered colimits. For this purpose it suffices to assume that trivial fibrations are closed under $\lambda$-filtered colimits. But I'd rather keep this answer less technical. | |
| Sep 29, 2013 at 10:42 | comment | added | Fernando Muro | Boris, is really necessary that the category be cofibrant? | |
| Sep 28, 2013 at 20:36 | history | answered | Boris Chorny | CC BY-SA 3.0 |