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 FT: A(T)→A and an induced representation T: D→A(T) such that F T T = 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. [ABSTRACT FROM AUTHOR]