Mixed Motives

A mixed Weil cohomology with values in an abelian rigid tensor category is a cohomological functor on Voevodsky’s category of motives which is satisfying Künneth formula and such that its restriction to Chow motives is a Weil cohomology. We show that the universal mixed Weil cohomology exists. Nori motives can be recovered as a universal enrichment of Betti cohomology via a localisation. This new picture is drawing some consequences with respect to the theory of mixed motives in arbitrary characteristic.