Timeline for Stabilization of a generic pointed model category
Current License: CC BY-SA 3.0
11 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 17, 2015 at 16:05 | vote | accept | Marc Nieper-Wißkirchen | ||
| Jul 17, 2015 at 5:36 | comment | added | David White | @MarcNieper-Wisskirchen: Thanks for the PS. This clarifies things for me. I only really ever think about cofibrantly generated situations, and I definitely only had that case in mind with the answer here. So I think functorial factorizations are no problem. Emily Riehl has done great work related to that topic, especially in her thesis. | |
| Jul 16, 2015 at 5:19 | comment | added | Marc Nieper-Wißkirchen | P.S.: Of course, my comment about non-functoriality applies only when trying to apply the construction to non-cofibrantly generated model categories. | |
| Jul 15, 2015 at 19:00 | comment | added | Zhen Lin | Yes, Quillen's original definition only calls for finite limits and finite colimits. But I also have my own reasons for thinking about small model categories and universe enlargement... | |
| Jul 15, 2015 at 17:14 | comment | added | David White | @DylanWilson: I did not say anything was wrong with it, and even if I thought there was something wrong this would have been the wrong forum to bring it up. I just said I can't follow it, and can't make it work for model categories. Am I right that the step reducing it to $C$ small is a universe enlargement, or is there something else going on? You are saying you can read the proof straight through and follow every step? If so I have questions for you. | |
| Jul 15, 2015 at 17:11 | comment | added | David White | @ZhenLin: I guess we both already thought about this point, and the conclusion was that you need to weaken the model category axioms to only ask for finite limits and colimits, right (i.e. take Quillen's version not Hovey's)? I'm thinking of mathoverflow.net/questions/106924 and mathoverflow.net/questions/108739 | |
| Jul 15, 2015 at 15:10 | comment | added | Marc Nieper-Wißkirchen | @DavidWhite: I'll go through the references you gave in the second to last paragraph. As to writing down the suspension functor via framings, at appears to me that one would need functorial factorizations to get functorial cosimplicial frames of cofibrant replacements. Without functoriality, I only see how to get a well-defined suspension functor on the homotopy category. So one may have to check how to extends Hovey's machinery to "functors, well-defined only up to homotopy". | |
| Jul 15, 2015 at 4:16 | comment | added | Dylan Wilson | There is no problem with 1.4.2.24 of Higher Algebra. The condition in the statement requires that $\mathcal{C}$ be pointed and have finite limits, which is a property clearly independent of the universe you're in. | |
| Jul 14, 2015 at 21:44 | comment | added | Zhen Lin | One can perfectly well have small model categories. There are no interesting small combinatorial model categories, however. | |
| Jul 14, 2015 at 20:34 | history | answered | David White | CC BY-SA 3.0 |