Abelian Hall categories

We will explain how, to any quiver, one can associate a finite length abelian category which categorifies the corresponding K-theoretic Hall algebra. The simples in this category provide a (dual) canonical basis of the Hall algebra. In particular, if the quiver is affine, this provides a basis for the positive half of the corresponding quantum toroidal algebra. We also explain how this abelian category is naturally endowed with renormalized r-matrices.