Timeline for Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 9, 2019 at 18:25 | vote | accept | Peter LeFanu Lumsdaine | ||
| Jan 9, 2019 at 16:44 | answer | added | Mike Shulman | timeline score: 1 | |
| Mar 19, 2018 at 15:56 | comment | added | Tyler Lawson | @DanGrayson Yes, I had forgotten this. Thanks. | |
| Mar 17, 2018 at 14:35 | comment | added | Dan Grayson | @TylerLawson -- can you not use Quillen's proof that the geometric realization of a Kan fibration is a Serre fibration? | |
| Mar 15, 2018 at 23:46 | answer | added | D.-C. Cisinski | timeline score: 2 | |
| Mar 15, 2018 at 14:45 | answer | added | Dan Grayson | timeline score: 2 | |
| S Oct 10, 2015 at 15:13 | history | bounty ended | CommunityBot | ||
| S Oct 10, 2015 at 15:13 | history | notice removed | CommunityBot | ||
| Oct 4, 2015 at 14:04 | comment | added | David White | It seems the answer for sSet is here in the comments. The key point is to use the Sing(|-|) adjunction to Top, where all objects are fibrant. The same trick should work for simplicial presheaves with the projective model structure, since fibrancy is detected levelwise. Peter, I think you should write an answer of your own using the comments and then if someone does better give them the bounty. Otherwise give yourself the bounty (if you can). I can't see how to do this in general. | |
| Oct 2, 2015 at 18:24 | comment | added | Karol Szumiło | Realizations of Kan fibrations are in fact (non-obviously) Serre fibrations. See e.g. 3.6.2 in Hovey's Model Categories. | |
| Oct 2, 2015 at 18:15 | comment | added | Tyler Lawson | Actually, I was not careful enough. A fibration $Y \to X$ of simplicial sets only realizes to a quasifibration, and $Sing$ applied to a quasifibration isn't necessarily a fibration -- so this won't fly. Sorry. | |
| Oct 2, 2015 at 15:22 | comment | added | Peter LeFanu Lumsdaine | @TylerLawson: oh… yes, that works, doesn’t it! Could you make that an answer? A generalisable answer would be even better, but that’s certainly good to be going on with… | |
| Oct 2, 2015 at 15:04 | comment | added | Tyler Lawson | I'd suggest using the standard non-fibrant construction of the fiberwise suspension, whose geometric realization is the standard fiberwise suspension in spaces (which preserves fibrations); then pull back the resulting map $Sing |\Sigma_X Y| \to Sing |X|$ along the map $X \to Sing |X|$, since applying the singular complex preserves fibrations. However, this doesn't really work in general model categories. | |
| Oct 2, 2015 at 14:11 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 3.0 |
edited to clarify as per Karol Szumiło’s comment
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| S Oct 2, 2015 at 14:07 | history | bounty started | Peter LeFanu Lumsdaine | ||
| S Oct 2, 2015 at 14:07 | history | notice added | Peter LeFanu Lumsdaine | Draw attention | |
| Oct 2, 2015 at 13:55 | comment | added | Peter LeFanu Lumsdaine | @KarolSzumiło: I only mean that factorisation in general can’t be stable under pb in the codomain, as witnessed by e.g. the inclusion of one point into a model of the circle with two distinct points. If we factor this as (TC,F), then every fibre of the resulting fibration is non-empty; whereas if we pull back along the inclusion of “the other point”, then factorise (i.e. fibrantly replace), we must get the empty space. But it seems very possible that for the restricted class of maps that occur as $\Sigma^{nf}_X Y \to X$, some hand-crafted factorisation could work. | |
| Oct 2, 2015 at 13:45 | answer | added | Jesse C. McKeown | timeline score: 4 | |
| Oct 2, 2015 at 9:35 | comment | added | Karol Szumiło | Could you elaborate on the "fairly obvious reasons" why factorizations cannot be stable? I think I have an idea for a carefully hand-crafted factorization, but the details would be messy... so before I try to check them I want to make sure that I'm not overlooking any obstacles. | |
| Oct 2, 2015 at 7:07 | comment | added | Peter LeFanu Lumsdaine | @JesseC.McKeown: in the slice, i.e. the map to X should be a fibration. Sorry if this was unclear. | |
| Oct 1, 2015 at 22:44 | comment | added | Jesse C. McKeown | Pardon me; in what category+ do you want the suspension to be fibrant? totalspace fibrant in SSet? fibration fibrant in SSet/X ? ... | |
| Sep 29, 2015 at 19:02 | history | edited | David White | CC BY-SA 3.0 |
edited title
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| Sep 29, 2015 at 18:06 | history | asked | Peter LeFanu Lumsdaine | CC BY-SA 3.0 |