Timeline for Homotopy equivalences preserving structure
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| May 9, 2013 at 11:46 | comment | added | Tom Goodwillie | When you say "for each nonempty intersection", it's not immediately clear whether by "nonempty" you mean that the intersection of the $X_{i_j}$ is nonempty or that $r>0$. But clearly (looking at your base case) you cannot mean the latter. And if the former then you seem to be forgetting that an empty intersection of $X_i$s might map to a nonempty intersection of the corresponding $Y_i$s. Why say "nonempty" at all? | |
| May 9, 2013 at 10:33 | answer | added | Ronnie Brown | timeline score: 2 | |
| May 9, 2013 at 9:30 | comment | added | Ricardo Andrade | @Oscar: Thank you for the clarification. I completely misunderstood the question. | |
| May 9, 2013 at 9:20 | answer | added | Andrew Ranicki | timeline score: 0 | |
| May 9, 2013 at 8:09 | comment | added | Oscar Randal-Williams | @Jeff, Ricardo: I don't think that is the question. I think the data being asked for is a map $g:Y \to X$ sending $Y_i$ into $X_i$ and homotopies $fg \to id_Y$ sending $Y_i$ into $Y_i$ for all time and $gf \to id_X$ sending $X_i$ into $X_i$ for all time. | |
| May 9, 2013 at 7:56 | history | edited | Benoît Kloeckner | CC BY-SA 3.0 |
LaTexed the question
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| May 9, 2013 at 3:59 | history | edited | Ricardo Andrade |
added tag
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| May 8, 2013 at 22:45 | comment | added | Ricardo Andrade | I do not know of any references for this particular fact. However, as Jeff Strom indicates, if you have "sufficient" homotopy extension properties, then the diagram formed by the intersections of the $X_i$'s (respectively, the $Y_i$'s) should be cofibrant, and thus its colimit is equivalent to its homotopy colimit. Then the homotopy invariance of homotopy colimits in the Hurewicz/Strøm model structure on topological spaces shows that the map induced on the colimits (which are hopefully $X$ and $Y$) is a homotopy equivalence. The book by Hirschhorn contains these model categorical facts. | |
| May 8, 2013 at 21:41 | comment | added | Jeff Strom | Can we express $X$ as a homotopy colimit of the various intersections and the map $f$ as a pointwise homotopy equivalence of diagrams? | |
| May 8, 2013 at 20:24 | history | asked | Steve Ferry | CC BY-SA 3.0 |