Monoidal abelian envelopes and a conjecture of Benson and Etingof

Abstract

We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. As a main result we prove that the category of finite-dimensional representations of a semisimple simply connected algebraic group is the abelian envelope of the category of tilting modules. Benson and Etingof conjectured that a certain limit of finite symmetric tensor categories is tensor equivalent to the finite-dimensional representations of ${\text{SL}}_2$ in characteristic $2$. We use our results on the abelian envelopes to prove this conjecture and its variants for any prime $p$.