A universal rigid abelian tensor category

TL;DR

This paper introduces a universal method to embed any rigid additive symmetric monoidal category into a rigid abelian one, providing new insights into longstanding conjectures in algebraic geometry.

Contribution

It presents a universal construction that connects rigid additive symmetric monoidal categories to rigid abelian categories, offering a new approach to key conjectures.

Findings

Established a universal embedding of categories

Provided a novel framework for Grothendieck's conjecture D

Linked to Voevodsky's smash nilpotence conjecture

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 yields a novel approach to Grothendieck's standard conjecture D and Voevodsky's smash nilpotence conjecture.