Timeline for What is the "universal problem" that motivates the definition of homotopy limits/colimits (and more generally "derived" functors)?
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 5, 2011 at 19:33 | vote | accept | Harry Gindi | ||
| Sep 10, 2010 at 17:31 | comment | added | Harry Gindi | @Bryan: Np | |
| Sep 10, 2010 at 15:22 | answer | added | Tom Goodwillie | timeline score: 4 | |
| Sep 9, 2010 at 4:00 | comment | added | Harry Gindi | @Sean: I will definitely be there (now that I know about it) since it's only an hour away! | |
| Sep 9, 2010 at 3:59 | comment | added | Harry Gindi | (this is because a left Quillen functor preserves weak equivalences between cofibrant objects, so precomposing it with a cofibrant replacement preserves all weak equivalences of C. Composing with the localization functor D->Ho(D) gives a functor C->Ho(D) sending all weak equivalences to isomorphisms, which must factor through the homotopy category. The unique induced morphism Ho(C)->Ho(D) is the "total left derived functor" that we can define using Kan extensions). | |
| Sep 9, 2010 at 0:15 | vote | accept | Harry Gindi | ||
| Feb 5, 2011 at 19:33 | |||||
| Sep 8, 2010 at 18:53 | comment | added | Harry Gindi | @Sam: Not the total derived functors, I mean the "left derived functor" of a left quillen functor C→D given by composition with the (for instance) cofibrant replacement, which induces the left total derived functor on homotopy categories by the universal property. | |
| Sep 8, 2010 at 18:50 | comment | added | Sean Tilson | Tyler is right, I was referring to the midwest topology seminar. | |
| Sep 8, 2010 at 18:42 | comment | added | Sam Isaacson | Harry, have you looked at Mike Shulman's paper "Homotopy limits and colimits and enriched homotopy theory"? In it, Shulman works out an enriched version of the homotopical machinery of Dwyer, Hirschhorn, Kan, and Smith; some things actually simplify in the enriched setting, since you can work explicitly with the spatial enrichment that is "secretly" there. Anyway, left and right derived functors from $C$ to $Ho(D)$ are right and left Kan extensions (respectively) along $C\to Ho(C)$. | |
| Sep 8, 2010 at 14:39 | answer | added | Dustin Mulcahey | timeline score: 7 | |
| Sep 8, 2010 at 13:25 | answer | added | Peter Arndt | timeline score: 16 | |
| Sep 8, 2010 at 12:38 | comment | added | Harry Gindi | To clarify a bit more, the motivation should seamlessly apply to the homotopical categories of Dwyer, Kan, Hirschhorn, and Smith, and the definitions for model categories should drop out from the nice properties enjoyed by model categories. | |
| Sep 8, 2010 at 12:35 | comment | added | Harry Gindi | I was looking mainly for formal properties. The other kind of motivation probably wouldn't help me anyway. | |
| Sep 8, 2010 at 12:32 | comment | added | Tyler Lawson | Also, it's not clear to me from the question whether the "motivation" you ask for is a motivation in terms of the formal properties enjoyed by ho(co)lims, or whether it's a question about the actual reasons they're employed (namely, what kinds of constructions this setup is supposed to abstract, collect together, and simplify). | |
| Sep 8, 2010 at 12:28 | comment | added | Tyler Lawson | Sean is referring to the Midwest Topology Seminar. | |
| Sep 8, 2010 at 12:26 | comment | added | Harry Gindi | Also, I don't understand what you're getting at. | |
| Sep 8, 2010 at 12:25 | comment | added | Harry Gindi | I got to the midwest on Sunday, and classes started yesterday, so don't worry so much! | |
| Sep 8, 2010 at 12:12 | comment | added | Sean Tilson | I can't help but wonder with all this topology you have been doing... will you make it to the midwest this fall? | |
| Sep 8, 2010 at 11:58 | history | asked | Harry Gindi | CC BY-SA 2.5 |