Timeline for When do colimits agree with homotopy colimits?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Apr 12, 2016 at 19:40 | vote | accept | Aly | ||
| Apr 12, 2016 at 18:27 | vote | accept | Aly | ||
| Apr 12, 2016 at 19:40 | |||||
| Apr 7, 2016 at 20:48 | comment | added | Karol Szumiło | @Dmitri That's a good observations, thanks for pointing that out. The two arguments don't seem that different though. The verification of the criterion you mention is essentially equivalent to the verification of the relevant properties of $\mathrm{Ex}^\infty$. That's the most laborious part of my argument, the rest is straightforward. In fact, you can even avoid using $\mathrm{Ex}^\infty$ altogether and use the fibrant replacement arising from the standard small object argument instead. It also preserves filtered colimits and is easier to establish than $\mathrm{Ex}^\infty$. | |
| Apr 7, 2016 at 16:15 | comment | added | Dmitri Pavlov | @KarolSzumiło: One can give a much shorter argument for the preservation of weak equivalences by any filtered colimit functor: using the classical sphere-filling criterion for weak equivalences of simplicial sets (see, for example, Proposition 4.1 in Dugger and Isaksen, Weak equivalences of simplicial presheaves), it suffices to observe that all simplicial sets involved in the criterion (i.e., Δ^n, ∂Δ^n, and their barycentric subdivisions) are compact and therefore factor through some finite stage. | |
| Apr 7, 2016 at 12:47 | comment | added | Karol Szumiło | BTW, I have noticed that the argument I gave is the same as the one in Charles Rezk's answer to the question you linked. | |
| Apr 7, 2016 at 12:43 | comment | added | Karol Szumiło | @Dmitri Thanks for the references. They are both very recent and are about much stronger results There should be some earlier references too, but I can't find any. The argument I gave above is also much more elementary. (And I find it quite amusing that we can prove homotopy invariance of filtered colimits by using fibrant replacements.) | |
| Apr 7, 2016 at 12:34 | comment | added | Dmitri Pavlov | @KarolSzumiło: For filtered colimits some references are given here: mathoverflow.net/questions/12746/… | |
| Apr 7, 2016 at 12:34 | comment | added | Karol Szumiło | @Dmitri You are absolutely right, in non-proper categories you need levelwise cofibrancy, but since the question was specifically about simplicial sets, I just skipped that remark. | |
| Apr 7, 2016 at 12:30 | comment | added | Dmitri Pavlov | @KarolSzumiło: There is a subtlety here: Reedy cofibrancy forces the objects a_i to be cofibrant. Pushouts along cofibrations in left proper model categories (which is what the OP mentioned) are always homotopy pushouts, even if a_i are not cofibrant. | |
| Apr 7, 2016 at 12:14 | comment | added | Karol Szumiło | @David I added some comments and references. | |
| Apr 7, 2016 at 12:14 | history | edited | Karol Szumiło | CC BY-SA 3.0 |
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| Apr 7, 2016 at 11:43 | comment | added | David White | These are great results. Can you provide a reference? Thanks! | |
| Apr 7, 2016 at 11:31 | history | answered | Karol Szumiło | CC BY-SA 3.0 |