Timeline for Model structure on category of endofunctors
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 3, 2013 at 12:23 | answer | added | David White | timeline score: 1 | |
| May 27, 2013 at 11:42 | comment | added | David White | @Shlomi: What ever happened to this? Did you figure out a way to make it work, or get whatever you needed from Chorny and Rosicky? You never told us what you wanted the model category for. Would an infinity category be enough? Check out page 50 of Lurie's HTT. He talks about putting objects and morphisms on the same footing. So an infinity category of infinity categories should have categories and functors on the same footing. Perhaps this is enough to "do homotopy" in End(C) | |
| Sep 12, 2012 at 19:28 | comment | added | Mike Shulman | @o a: But not, perhaps "non-trivial" in the less technical sense of "interesting"... (-: | |
| Sep 12, 2012 at 5:53 | comment | added | o a | David White: not quite, a small posetal category may all small limits and colimits, and may have a non-trivial model category structure. (Non-trivial in the technical sense that it is not one of the three trivial model category structures where one of the three classes is the class of all morphisms). | |
| Sep 11, 2012 at 20:28 | comment | added | Shlomi A | Thanks David. I'll take a look into that paper. I'm interested in the category of all endofunctors, however. I have the feeling that in order to have all small (co)limits in $End(\mathcal C)$, one cannot take less then that. | |
| Sep 11, 2012 at 17:25 | comment | added | David White | You might be interested in the following paper of Chorny and Rosicky. They extend the notion of a combinatorial model category and as a consequence find all sorts of new model categories. In particular, they prove that the category of small presheaves over large indexing category is a model category. Perhaps the methods used there will help. arxiv.org/abs/1110.4252. Here's a sketch of why I think it might help. If $C$ is cofibrantly generated, then maybe Quillen endofunctors are determined by where the generators go, so act like small presheaves. Not sure about all endofunctors, though. | |
| Sep 11, 2012 at 17:05 | comment | added | David White | It's a good thing you're not assuming $\mathcal{C}$ is small, because there are no nontrivial small model categories. This comment appears on page 15 of Hovey's book, but is simply a consequence of the fact that model categories must have all small limits and colimits | |
| Sep 11, 2012 at 15:33 | history | asked | Shlomi A | CC BY-SA 3.0 |