# How much of the category of motives can be recovered from automorphisms of the Betti functor

Say we are working with schemes over a field $$k\subset \mathbb{C}.$$ A motive in the sense of Voevodsky is a functor $$Sch\to D^bVect$$ from (an appropriate category of) schemes to the DG category of complexes of vector spaces satisfying some descent and locality conditions such as $$A^1$$-locality (I will be purposefully ambiguous about the specific conditions, as I don't want to commit to a specific motivic category). Let $$M$$ be the category of such motives. Then $$M$$ is a full subcategory of the functor category $$Fun(Sch, D^bVect).$$ Now certain motivic categories are known to be Tannakian or derived Tannakian, i.e. fully controlled (as a symmetric monoidal category) by an algebraic group $$G$$, which is defined to be the category of automorphisms of a (symmetric monoidal) fiber functor $$F_M:M\to D^bVect.$$ If the fiber functor $$F$$ is "sufficiently nice" then it is representable in the functor category by a functor $$F:Sch\to D^bVect,$$ and thus its (algebraic) group of automorphisms is equivalent to the group of automorphisms of the functor $$F.$$

As the notion of fiber functor is quite flexible, this seems like a very promising approach to constructing motivic categories: namely, take a "nice" symmetric monoidal functor $$F:Sch\to D^bVect,$$ consider its algebraic group of (symmetric monoidal) automorphisms, $$G$$, and take the category of representations of $$G$$.

I haven't seen a careful construction of motivic invariants from this point of view, but it seems very plausible to me that such a description exists. For example on the level of Tate motives, the associated graded of the weight filtration is known to be a fiber functor, which seems to imply that automorphisms of the functor $$X\mapsto Gr(C^*(X))$$ (graded with respect to the weight filtration), which should be noncanonically equivalent to the Betti homology functor, should have automorhpism group which completely controls mixed Tate motives. Perhaps something similar can be said with the full motivic category and the algebraic $$K$$ group functor.

Is something like this true? If true, this would imply that motivic Galois groups can be defined without the category of motives, as automorhpisms of a a suitable "generating" functor. Is this written up somewhere?

• For a related construction, see Section 2.5 (especially the paragraphs immediately before Example 2.17) of arxiv.org/abs/1812.10822 and Example 2.20 of arxiv.org/abs/1309.0637 . – Jon Pridham Oct 13 '19 at 7:55
• I would recommend you to read Ayoub and also (Huber's book?) on Nori motives. – Mikhail Bondarko Oct 13 '19 at 15:49
• Thank you. I had not read Ayoub's papers and they seem to be very close to what I want. And Jon -- I've seen your papers before, but didn't realize their motivic context - thanks. I also found the answer to this question to be helpful: mathoverflow.net/questions/254341/… – Dmitry Vaintrob Oct 17 '19 at 2:25