Timeline for Grothendieck derivators vs $\infty$-categories
Current License: CC BY-SA 4.0
22 events
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| Jun 10, 2021 at 9:39 | comment | added | Fernando Muro | Excellent point @UrsSchreiber No, I don't know of any reference but it looks like something interesting to look at. I actually remember someone rising this question last time I talked about derivators a couple of years ago, in person. | |
| Jun 10, 2021 at 7:26 | comment | added | Urs Schreiber | @FernandoMuro I think your pointer to Renaudin's result ncatlab.org/nlab/show/Ho(CombModCat) is the most pertinent reply here, since the only way to really know how derivators relate to ∞-categories is to compare the (large) categories (n-categories) which both form. The only trouble is that your comment takes for granted a fact whose proof seems to be sadly missing in the literature: That the 2-category of combinatorial model categories localized at Quillen equivalences is equivalent to the homotopy 2-category of presentable ∞-categories. This ought to be true, but is there a proof? | |
| S Aug 3, 2020 at 3:18 | history | bounty ended | Amos Kaminski | ||
| S Aug 3, 2020 at 3:18 | history | notice removed | Amos Kaminski | ||
| Aug 1, 2020 at 17:58 | vote | accept | Amos Kaminski | ||
| Jul 30, 2020 at 11:34 | comment | added | Fernando Muro | My second comment is kind of infamous, one has to put a lot of will to understand it well, I'm tempted to erase it. | |
| Jul 30, 2020 at 11:31 | comment | added | Fernando Muro | @mikeshulman I actually said "very roughly" ;) thanks for stressing this part, it's difficult to sum up, better to look at the reference, which doesn't even talk about infinity categories! | |
| Jul 30, 2020 at 11:29 | comment | added | Mike Shulman | It took me a while to figure out what you meant by "derivators are to ∞-categories what homotopy theory is to topology". Would a more precise version of the statement end with "...what 𝛱-algebras are to spaces"? | |
| Jul 30, 2020 at 11:22 | comment | added | Mike Shulman | @FernandoMuro Your statement is missing the very important adjective "locally presentable" in front of "derivators" and "$(\infty,1)$-categories". (I know you said "roughly speaking", but I think this is important not to omit. For one thing, a derivator is always complete and cocomplete, whereas an arbitrary $(\infty,1)$-category is not!) | |
| Jul 29, 2020 at 18:04 | comment | added | Fernando Muro | So, yes, you miss information but you do keep a lot. Also, very roughly speaking (and shifting dimensions by $-1$), derivators are to $\infty$-categories what homotopy theory is to topology. So, if you like homotopy theory, probably you should also like derivators as much. | |
| Jul 29, 2020 at 18:01 | comment | added | Fernando Muro | Very roughly speaking, by [Ren09] the $2$-category of derivators is equivalent to: 1. Take the $(\infty,1)$-category of $(\infty,1)$-categories. 2. Truncate it to a $(2,1)$-category. 3. Perform a $2$-categorical localization inverting those $1$-morphisms which induce an equivalence on homotopy categories. [Ren09] Renaudin, Olivier. 2009. “Plongement de Certaines Théories Homotopiques de Quillen Dans Les Dérivateurs.” Journal of Pure and Applied Algebra 213 (10): 1916–1935. doi.org/10.1016/j.jpaa.2009.02.014. | |
| Jul 29, 2020 at 17:37 | answer | added | Mike Shulman | timeline score: 26 | |
| S Jul 29, 2020 at 12:20 | history | bounty started | Amos Kaminski | ||
| S Jul 29, 2020 at 12:20 | history | notice added | Amos Kaminski | Draw attention | |
| Jul 29, 2020 at 9:20 | comment | added | Mirco A. Mannucci | Deleted my answer. Now, Kevin, Harry and Mike, I would be very interested in reading your answers here. | |
| Jul 28, 2020 at 6:33 | comment | added | Kevin Carlson | I don't know that I have a really compelling answer here, but to Denis and Lennart's points the answer is: at best only in terms of homotopy limits in a model structure, which is a major advantage of $\infty$-categories. Derivators are better suited to working within a single homotopy theory at a time. Regarding your second question, well, since Lurie began writing there has been vastly more machinery developed for $\infty$-categories, and some things (see above) have been done only in that framework. | |
| Jul 27, 2020 at 17:46 | comment | added | Denis Nardin | Another version of Lennart's comment: can you talk about sheaves of derivators? One of the great strenghts of ∞-cats is that they work very well in families (so, for example, you can rephrase faithfully flat descent as "$\mathrm{QCoh}(-)$ is a sheaf") | |
| Jul 27, 2020 at 17:19 | history | edited | Amos Kaminski | CC BY-SA 4.0 |
added 20 characters in body
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| Jul 27, 2020 at 16:59 | comment | added | Fernando Muro | The following paper shows interesting relations between both: Arlin, Kevin. 2020. “On the $\infty$-Categorical Whitehead Theorem and the Embedding of Quasicategories in Prederivators.” ArXiv:1612.06980 [Math], February. arxiv.org/abs/1612.06980. | |
| Jul 27, 2020 at 11:11 | comment | added | Lennart Meier | Can one take a limit of a diagram of derivators? | |
| Jul 27, 2020 at 9:56 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Grammar and tags
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| Jul 27, 2020 at 9:10 | history | asked | Amos Kaminski | CC BY-SA 4.0 |