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

Question

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?

Answer 1

A precise way to say that "sufficiently nice functors" from schemes to complexes of vector spaces determine realization functors of Voevodsky's motives is the notion of mixed Weil cohomology.

There are rather explicit descriptions (i.e. categories) which describe what can be determined by the automorphisms of Betti cohomology. For the derived version, you might have a look at:

I. Iwanari, Tannakization in derived algebraic geometry, J. K-Theory 14 (2014), 642—700.

(also https://arxiv.org/abs/1112.1761).

The non derived (hence coarser) version is the theory of Nori motives (introduced by Nori in the early 2000's), which is the subject of a recent book of A. Huber and of S. Müller-Stach, published by Springer in 2017. Yves André's Bourbaki talk is a good source as well, in which Ayoub's contributions are well explained.

Either way, none of these constructions determines Voevodsky's theory of motives. It is not known that the Betti realization functor is conservative on Voevodsky's motives (Ayoub is working hard on it for years, but there is no complete proof so far). Morever, the conservativity of the Betti realization would not be sufficient to prove that Voevodsky's motives form a (derived) tannakian category (conservativity only implies that the tannakian category associated to Voevodsky's motives does not depend on the choice of a realization functor such as Betti cohomology). But more optimistic conjectures would imply that. To be more precise: the approach of Voevodsky has two important features: it satisfies a reasonable universal property (in some higher categorical sense) and the main cohomology theory represented there (i.e. determined by the unit of the tensor product) is classical intersection theory (under the form of (higher) Chow groups). This latter property is what is lost (until we prove very non-trivial conjectures which go much beyond conservativity, such as the Hodge conjecture) when we go to more tannakian versions of motives.

Finally, this idea of defining motives by forcing Betti cohomology to be a fiber functor is not new at all: if we restrict to pure motives (motives of smooth and projective varieties), this boils down to Deligne's theory of `cycles motivés' revisited by Yves André in 1996 (this is very well explained in terms of Nori's approach at the end of the book of Huber and Müller-Stach).