# Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)

Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G \to \mathcal C$. Assume every object of $\mathcal C$ is a summand of an object in the image of this functor $Rep_G \to \mathcal C$.

To check that this functor $Rep_G \to \mathcal C$ is an equivalence, it is sufficient to check that it is fully faithful. (Essential surjectivity follows from fullness and the summand condition.)

Is it sufficient to check the fully faithfulness condition on finitely many objects of $Rep_G$?

In my previous question, with an additional assumption about a fibre functor, Ehud Meier showed that $\mathcal C$ must be symmetric and thus by the Tannakian correspondence $\mathcal C$ must be the representation category of a subgroup of $G$. By some group theory (Larsen's alternative + Goursat's Lemma) I can then find a finite set of objects to check fully faithfulness on. But what if I don't have a fibre functor? Surely $\mathcal C$ need not always be symmetric? Is it still true?

• Your point being that there is at most one symmetry on $\mathcal C$ such that the functor is symmetric monoidal, but in fact there need not be any such symmetry? – Theo Johnson-Freyd Feb 6 '16 at 16:59
• @TheoJohnson-Freyd Yes, there need not be any such symmetry (as far as I know), but if it exists then it is unique. – Will Sawin Feb 6 '16 at 20:50