Definable categories and $\mathbb{T}$-motives

Abstract

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