Definable categories and T-motives

TL;DR

This paper constructs a universal abelian category for quiver representations in abelian categories, linking T-motives and Nori's motives to functor categories on definable categories.

Contribution

It introduces a universal abelian category framework for quiver representations, connecting motives with definable categories and functor categories.

Findings

Existence of a universal abelian category for quiver representations.

Representation of T-motives and Nori's motives via functor categories.

Establishment of a universal property for the constructed category.

Abstract

Making use of Freyd's free abelian category on a preadditive category we show that if $T:D\rightarrow \mathcal{A}$ 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.