Timeline for If a colimit of distinguished triangles exists, is it also a distinguished triangle?
Current License: CC BY-SA 2.5
12 events
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| Aug 25, 2010 at 17:04 | comment | added | Sean Tilson | This happens to be the only triangulated category I am familiar with, and I actually need this in some sense which is why I was curious about your objections. It seems that, to me, you are objecting because of phantom map type phenomena, which we do not have for (homotopy) colimits. If I am wrong about any of this, please let me know. | |
| Aug 25, 2010 at 17:01 | comment | added | Sean Tilson | sure thing. I was only using the fact that Cofibrations can be thought of as homotopy colimits to morally argue for my point. This might not be the case in general, but if we are looking at the stable homotopy category of the category of spectra then I am pretty sure that my argument works. Also, this is not a homotopy limit, this is a homotopy colimit, so maybe this is part of the reason we don't have to worry so much. | |
| Aug 25, 2010 at 16:18 | comment | added | Mikhail Bondarko | Actually, I am definitely not a specialist in this topic. I am just trying to say that when you have a morphism of homotopy limits, it possibly does not have to come from a morphism of the corresponding system of spaces. | |
| Aug 18, 2010 at 20:29 | comment | added | Sean Tilson | I guess you are right earlier when you say that I am only thinking of maps in the homotopy category that come from maps of spaces. Maybe I am being dumb, but I did not know that there were other maps. perhaps this is the type of thing we should discuss via email? stilson AT wayne DOT edu | |
| Aug 17, 2010 at 18:42 | comment | added | Mikhail Bondarko | It seems that one alraedy has problems when considering homotopy classes of maps (instead of maps themselves). | |
| Aug 16, 2010 at 15:07 | comment | added | Sean Tilson | I am confused, maybe i don't understand what the homotopy category is. I was thinking about working in the category where objects are spaces (CGH etc etc) and morphisms are homotopy classes of continuous maps. I would even be willing to restrict to CW complexes. Are the things you are thinking about zig-zags? | |
| Aug 9, 2010 at 9:56 | comment | added | Mikhail Bondarko | I think that in your answer you only consider morphisms of inductive systems in the homotopy category that could be lifted to the level of spaces. For a morphism of inductive systems that does not possess such a lift, the answer is probably "no" in general. | |
| Jul 26, 2010 at 17:23 | comment | added | Sean Tilson | Tom is one of the obvious parties... | |
| Jul 26, 2010 at 17:23 | comment | added | Sean Tilson | i would imagine that homotopy colimits better commute with homotopy colimits. if they don't then we are in trouble. have you looked at any dugger-isaksen stuff? i think it might be the case that in a triangulate model category the statement you ask about is in fact true, so that would pass through to your setting. However, all this is speculation that should be cleared up by someone more knowledgeable, if one of the obvious parties doesnt check in before i get a response from someone i will post it in my answer. | |
| Jul 26, 2010 at 9:06 | comment | added | Andreas Holmstrom | Maybe an argument in this spirit could work, but one question would be the following: Does homotopy colimits commute with homotopy colimits in general? | |
| Jul 26, 2010 at 7:08 | comment | added | Andreas Holmstrom | Yes, I'm interested in the motivic stable homotopy category over some base scheme. Most statements that hold in the topological setting should hold there as well. | |
| Jul 26, 2010 at 5:53 | history | answered | Sean Tilson | CC BY-SA 2.5 |