Timeline for fibre-preserving homotopy equivalence
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 3, 2022 at 7:43 | comment | added | Ken | This also follows from formality: Give $\mathsf{Top}$ with the Hurewicz model structure. The arrow category $\operatorname{Arr}(\mathsf{Top})$ admits the injective model structure. In this model structure, every object is cofibrant; fibrant objects are precisely the fibrations in $\mathsf{Top}$. If $f$ and $\overline{f}$ are homotopy equivalences, then $(f,\overline{f})$ is a weak equivalence in $\operatorname{Arr}(\mathsf{Top})$ between fibrant-cofibrant objects, and hence is a homotopy equivalence. | |
| Jul 1, 2021 at 19:27 | vote | accept | Mark Grant | ||
| Jul 1, 2021 at 19:27 | answer | added | Mark Grant | timeline score: 1 | |
| Jun 4, 2021 at 16:46 | comment | added | Mark Grant | @GustavoGranja well I didn't check all details, but enough to have faith that it's true. If you'd like to post this reference as an answer, I'd gladly accept it. | |
| Jun 3, 2021 at 14:09 | comment | added | Mark Grant | @GustavoGranja It would appear that Peter May thinks so: see p. 53 of the same book! I'll try to check the details and get back to you. Barring a major surprise, this reference seems to be a satisfactory answer. | |
| Jun 3, 2021 at 12:58 | comment | added | Gustavo Granja | I don't think so. Doesn't the proof of the Proposition on p. 47 of Peter May's book dualize? | |
| Jun 1, 2021 at 13:23 | history | asked | Mark Grant | CC BY-SA 4.0 |