This is a rather imprecise question but i think this could become a interesting pool of ideas and comments.

The theory of motives has evolved to a complex field of research the moment Voevodsky (and others) have introduced the triangulated categories and concepts of homotopy theory to the field. The/one motivation behind (stable) derivators is to enable a functorial cone construction. This is a problem arising in the theory of triangulated categories.

What improvements in the theory of motives could be derived from the theory of derivators?

Noticing that both concepts where introduced by late Grothendieck made me think if maybe there originally was a conceptual link between these topics in the first place, when he came up with the derivators.

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    $\begingroup$ 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. $\endgroup$
    – AAK
    Nov 17, 2014 at 8:34
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    $\begingroup$ 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). $\endgroup$
    – AAK
    Nov 17, 2014 at 8:36
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    $\begingroup$ [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/… $\endgroup$ Sep 17, 2017 at 3:21
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    $\begingroup$ 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 $\endgroup$ Sep 17, 2017 at 10:34
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    $\begingroup$ @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). $\endgroup$ Sep 19, 2017 at 13:56


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