We introduce a general formalism allowing one to associate to some monoidal functors a Hopf algebra in the target category. This formalism is applied to the Betti realization of Voevodsky's motives over a base field k endowed with a complex embedding. We obtain in this way a Hopf algebra in the derived category of Q-vector spaces. Using the comparison theorem of singular cohomology with algebraic de Rham cohomology, we deduce that this Hopf algebra has no homology in strictly negative degrees. Its zero degree homology is thus a Hopf algebra in the usual sense and its spectrum is called the motivic Galois group. We study different aspects of these motivic Hopf algebras and these motivic Galois groups.
Joseph Ayoub <firstname.lastname@example.org>