Timeline for When are (weak) homotopy equivalence testable on open covers?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| May 1, 2015 at 18:15 | vote | accept | Tom | ||
| Mar 24, 2015 at 3:20 | comment | added | Lennart Meier | The notion of a homotopy colimit clearly depends both on the category and the chosen class of weak equivalences. Tom Diecks definition of homotopy colimits is in the same style as the usual textbook definition of (co)homology theories, without the weak equivalence axiom, so that one can include things like Cech cohomology. | |
| Mar 23, 2015 at 13:24 | comment | added | Dmitri Pavlov | But not for a completely standard notion such as a homotopy colimit. Should the reader also be blamed if the author uses a nonstandard definition of a topological space (say)? | |
| Mar 23, 2015 at 12:35 | comment | added | Karol Szumiło | The way I see it, readers who skip definitions can only blame themselves. | |
| Mar 23, 2015 at 12:30 | comment | added | Dmitri Pavlov | Yes, tom Dieck does define it, but “clarifier” really means “adjective” above, i.e., somebody who just reads this particular statement may be misled if he doesn't also read the definition above. | |
| Mar 23, 2015 at 12:27 | comment | added | Karol Szumiło | That terminology is indeed non-standard, but it is unfair to say that it is not clarified. The definition is given right above Proposition 4.2.3 which I originally cited. (Tom later changed this to 4.2.7 which does answer his question more directly.) | |
| Mar 23, 2015 at 12:26 | vote | accept | Tom | ||
| May 1, 2015 at 18:15 | |||||
| Mar 23, 2015 at 12:22 | comment | added | Dmitri Pavlov | Indeed, thanks for the clarification. I must say that the usage of “homotopy pushout” in tom Dieck's book without any additional clarifiers to mean “homotopy pushout in topological spaces equipped with homotopy equivalences” is rather confusing. | |
| S Mar 23, 2015 at 11:21 | history | suggested | Tom | CC BY-SA 3.0 |
The theorem is better suited to the question.
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| Mar 23, 2015 at 11:01 | review | Suggested edits | |||
| S Mar 23, 2015 at 11:21 | |||||
| Mar 23, 2015 at 11:00 | comment | added | Karol Szumiło | @Dmitri That's true for homotopy pushouts with respect to weak homotopy equivalences and that's what the first part of my answer says. The notion of a homotopy pushout with respect to homotopy equivalences is stronger and here you need numerability. | |
| Mar 23, 2015 at 10:43 | comment | added | Dmitri Pavlov | The original post says that U and V are open, and in this case the pushout is always a homotopy pushout, regardless of numerability. | |
| Mar 23, 2015 at 10:36 | history | edited | Karol Szumiło | CC BY-SA 3.0 |
added 2 characters in body
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| Mar 23, 2015 at 10:27 | history | answered | Karol Szumiło | CC BY-SA 3.0 |