Timeline for Use of derivators to the theory of motives?
Current License: CC BY-SA 3.0
20 events
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| S Sep 25, 2017 at 4:46 | history | bounty ended | CommunityBot | ||
| S Sep 25, 2017 at 4:46 | history | notice removed | CommunityBot | ||
| Sep 19, 2017 at 13:56 | comment | added | Dmitri Pavlov | @DylanWilson: I am pretty sure the answer is no. This was meant to be a comment on your reply to Adeel's remark that ∞-categories are "stronger" refinements than derivators (they indeed are). | |
| Sep 19, 2017 at 10:49 | comment | added | Dylan Wilson | I dunno- I've never used derivators for anything myself, I just happen to know that fun fact. Maybe you should ask a separate MO question about it? | |
| Sep 19, 2017 at 3:50 | comment | added | Dmitri Pavlov | @DylanWilson: Can you recover the ∞-category of left adjoint functors between two presentable stable ∞-categories from the corresponding category of morphisms of their associated derivators? Can you compute the pullback of a diagram of presentable stable ∞-categories if you are given the associated diagram of derivators? (The latter example actually arose in my work in a nonabstract context.) | |
| S Sep 17, 2017 at 22:50 | history | suggested | user111474 |
Added Homotopy tag
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| Sep 17, 2017 at 21:52 | review | Suggested edits | |||
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| Sep 17, 2017 at 10:34 | comment | added | David Roberts♦ | What Dylan meant was this doi.org/10.1016/j.jpaa.2009.02.014, which is Plongement de certaines théories homotopiques de Quillen dans les dérivateurs, by Olivier Renaudin | |
| Sep 17, 2017 at 9:34 | comment | added | Leo Alonso | @DylanWilson The link does not work outside of Northwestern. | |
| Sep 17, 2017 at 3:21 | comment | added | Dylan Wilson | [This is unrelated but wanted to point out that, contrary to Adeel's claim, one can recover a (presentable) stable infty-category from its associated derivator. (see sciencedirect.com.turing.library.northwestern.edu/science/… | |
| S Sep 17, 2017 at 2:48 | history | bounty started | tttbase | ||
| S Sep 17, 2017 at 2:48 | history | notice added | tttbase | Draw attention | |
| Nov 19, 2014 at 15:57 | comment | added | Simon Pepin Lehalleur | Two comments on algebraic derivators: 1) the work of Ayoub on the Betti realisation and Ayoub and Cisinski-Deglise on the l-adic realisation shows that the classical theories of systems of coefficients (derived categories of topological sheaves for algebraic varieties over \mathbb{C} and Ekedahl's derived categories of l-adic sheaves) form algebraic derivators and 2) the language of algebraic derivators is useful to express various subtle geometric situations: cohomological descent, stratifications (see Ayoub-Zucker), nearby cycles functor. | |
| Nov 17, 2014 at 15:37 | comment | added | nxir | Thanks so far, Adeel. I had no idea of algebraic Derivators. I will have a look on these | |
| Nov 17, 2014 at 13:56 | comment | added | Mikhail Bondarko | I would like to express my own understanding of Adeel's answer. So, when treating motives over a base one considers certain triangulated motivic categories over each base scheme and also several types of connecting functors between these categories. The motivic categories are equipped with canonical models, and some of the connecting functors possess canonical lifts to models; yet some of the types of these functors do not seem to possess canonical lifts of this sort. I would really like to know whether algebraic derivators solve this problem! | |
| S Nov 17, 2014 at 9:53 | history | suggested | AAK |
more appropriate tags
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| Nov 17, 2014 at 9:28 | review | Suggested edits | |||
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| Nov 17, 2014 at 8:36 | comment | added | AAK | However, a beautiful idea of Ayoub is to modify the definition of derivator, replacing the 2-category of diagrams by the 2-category of diagrams of schemes over some base, and imposing various axioms. The resulting notion, which he calls an algebraic derivator, essentially captures the whole formalism of six functors. I am not sure, but I believe that Grothendieck probably did not have anything like this in mind when he wrote Pursuing Stacks or Les Derivateurs (and I will be very happy if an expert could confirm). | |
| Nov 17, 2014 at 8:34 | comment | added | AAK | In the motivic world, model categories and infinity-categories are used as refinements of triangulated categories. These are "stronger" refinements than derivators (in some precise sense), so I doubt that the theory of derivators can offer any improvements to the theory of motives. | |
| Nov 17, 2014 at 0:41 | history | asked | nxir | CC BY-SA 3.0 |