Timeline for Homotopy Equivalence of Mapping Cones
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 11, 2013 at 15:57 | comment | added | Ronnie Brown | @Fernando: All I am suggesting is that my argument for the gluing lemma, which was first in print in 1968 in my book, can presumably be translated into a model category with a cylinder object, and should give explicit maps and homotopies in the case asked. | |
| May 28, 2013 at 9:10 | comment | added | Fernando Muro | @Ronnie, I thing that model category arguments are usually very explicit about homotopies. | |
| May 14, 2013 at 6:53 | comment | added | Ronnie Brown | Model category arguments, and I presume also for triangulated categories, tend to be non explicit on homotopies. So I would expect my answer to mathoverflow.net/questions/130116 could be relevant, since it gives explicit desription of homotopies in an analogous situation, relevant to, indeed used as a basis for a basis for, gluing homotopy equivalances. | |
| Jan 22, 2013 at 2:38 | comment | added | Marc Hoyois | You indeed only get a canonically induced map of cones once you choose the homotopy $\psi_1$. That's why taking cones is not a functorial construction in a triangulated category. To make the cone construction functorial you must consider the map $\psi_1$ as part of the data defining a commutative square, which is what higher category theory does. | |
| Jan 21, 2013 at 23:28 | comment | added | user30838 | Up to choice of homotopy, it's pretty clear how to construct the maps from looking at the induced homotopy exact squares. I'm specifically having issues with the fact that for us to have a homotopy equivalence of cones that the choice of homotopy is important (or I'm missing something completely). I'll go back and think about the proof more and thanks! | |
| Jan 21, 2013 at 20:55 | comment | added | Fernando Muro | Constructing explicitly the map may be a mess, but it is completely elementary. An excersise, in order to fully understand why the homotopy category is triangulated. | |
| Jan 21, 2013 at 20:49 | history | asked | Mike | CC BY-SA 3.0 |