A universal rigid abelian tensor category

Abstract

We prove that any rigid additive symmetric monoidal category can be mapped to a rigid abelian symmetric monoidal category in a universal way. This sheds a new light on abelian $\otimes$-envelopes and on motivic conjectures such as Grothendieck's standard conjecture $\mathrm{D}$ and Voevodsky's smash nilpotence conjecture.