  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 $SL_2$ in characteristic $2$. We use our results on the
abelian envelopes to prove this conjecture and its variants for any prime $p$.
