Definable categories and T-motives

Making use of Freyd’s free abelian category on a preadditive category we show that if T:D → A is a representation of a quiver D in an abelian category A then there is an abelian category A(T), a faithful exact functor F_T:A(T) → A and an induced representation T^:D → A(T) such that F_TT^ = T universally. We then can show that T-motives as well as Nori’s motives are given by a certain category of functors on definable categories.