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 3.0
17 events
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| Oct 13, 2015 at 17:22 | history | edited | Leo Alonso | CC BY-SA 3.0 |
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| Sep 10, 2010 at 14:18 | history | edited | Peter Arndt | CC BY-SA 2.5 |
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| Sep 10, 2010 at 12:07 | comment | added | Peter Arndt | Ah, got it: It is $Q(i_D)$, as the last arrow in the last zig-zag that would make the square commute telling us that the whole thing is $\tau′_D$, but $i_{QD}$ that goes into the naturality square of the transformation $colim \circ Q \rightarrow colim$... | |
| Sep 9, 2010 at 16:05 | comment | added | Tom Goodwillie | Peter, I think that even to get the homotopy-category statement you need one more step: The two available maps $QQD\to QD$ are not a priori equal, even though are both equivalences. To see that they are equal you have to use their compositions with the three maps $QQQD\to QQD$. | |
| Sep 9, 2010 at 15:43 | comment | added | Peter Arndt | I added something about this as well, but shied away from proving the whole homotopical statement :-) | |
| Sep 9, 2010 at 15:41 | history | edited | Peter Arndt | CC BY-SA 2.5 |
proved more
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| Sep 9, 2010 at 1:55 | comment | added | Tom Goodwillie | I mean, of the map from F to colimQ (in the category that you get from the category of all homotopy-invariant functors over F by inverting the weak equivalences). | |
| Sep 9, 2010 at 1:08 | comment | added | Tom Goodwillie | To show that colimQ is homotopically terminal you must establish uniqueness (in an appropriate sense) as well as existence of the map from FD. But this can be done, too. | |
| Sep 9, 2010 at 0:15 | vote | accept | Harry Gindi | ||
| Feb 5, 2011 at 19:33 | |||||
| Sep 9, 2010 at 0:15 | comment | added | Harry Gindi | I'm following the terminology of Dwyer, Hirschhorn, Kan, and Smith, but this answers the question just fine. Thanks. | |
| Sep 8, 2010 at 23:40 | comment | added | Peter Arndt | I added that to the answer, didn't fit into the comment. I hope it's what you wanted to know - I am not sure about your terminology... | |
| Sep 8, 2010 at 23:37 | history | edited | Peter Arndt | CC BY-SA 2.5 |
added proof requested in a comment
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| Sep 8, 2010 at 18:13 | comment | added | Harry Gindi | Peter, how does one prove that $hocolim:=colim(Q(-))$ is homotopically terminal in the "homotopical category of homotopical functors over colim"? | |
| Sep 8, 2010 at 18:12 | history | edited | Peter Arndt | CC BY-SA 2.5 |
cleaned typos and grammar
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| Sep 8, 2010 at 15:26 | comment | added | Tim Porter | Nice answer. I like the situation in simplicially enriched categories which are complete or cocomplete enough. There you can actually read off a universal homotopy coherent cone or cocone so as to visualise the universal property. The (co)simplicial replacement business of Bousfield and Kan is also a good way to gain an intuition of the holim or hocolim construction. Of course, this works best with weak equivalences being homotopy equivalences and so your point about fibrant and cofibrant objects is important. | |
| Sep 8, 2010 at 13:30 | history | edited | Peter Arndt | CC BY-SA 2.5 |
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| Sep 8, 2010 at 13:25 | history | answered | Peter Arndt | CC BY-SA 2.5 |