Timeline for Homotopy fibre sequence and left Bousfield localization
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 15, 2019 at 16:54 | history | edited | David White | CC BY-SA 4.0 |
Fixed typo on this new question, and added top level tag
|
| Sep 15, 2019 at 16:54 | answer | added | David White | timeline score: 5 | |
| Sep 14, 2019 at 21:09 | comment | added | Lao-tzu | What you said is true, that follows from the fibrewise characterization of being homotopy cartesian in sSet. There is indeed a five lemma type result in Hovey's book: (dual of ) Proposition 6.5.3. But that is about the fiber. | |
| Sep 14, 2019 at 21:03 | comment | added | Kevin Carlson | I do think the result needs to be modified, since certainly a map of fibrations of pointed but not necessarily connected simplicial sets which is an equivalence on base and fiber has no reason to be an equivalence on the total space. But I believe the analogous statement is true for a map of fibrations of non-pointed simplicial sets which is an equivalence on the base and on every choice of fiber. | |
| Sep 14, 2019 at 20:55 | comment | added | Lao-tzu | You are right. Unfortunately, my main concern is the non-stable case. I guess mostly probably the answer is negative, as in your comparing of fiber sequences (I think it comes down to result about sSet). | |
| Sep 14, 2019 at 20:51 | comment | added | Kevin Carlson | If $\mathcal M$ is stable, then comparing the fiber sequences $Map(X,F)\to Map(X,E)\to Map(X,B)$ to $Map(A,F)\to Map(A,E)\to Map(A,B)$ for any $A\to X$ in $\mathcal C$, one gets the desired result from the five-lemma. Otherwise, I'm not sure what happens; a bit of digging around found no clear statement of a five lemma type result for maps of fibrations of spaces or pointed spaces which are not necessarily connected, as in this case the mapping spaces cannot possibly be. | |
| Sep 14, 2019 at 14:59 | history | asked | Lao-tzu | CC BY-SA 4.0 |