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I wonder the state of the following conjecture in "Deformation theory, Kontsevich, Soibelman":

Conjecture 3.3.5. Rigid [abelian symmetric] tensor categories [over an algebraically closed field $k$] with ranks in $\mathbb{Z}$ can be of two types: comodules over commutative Hopf algebras or comodules over supercommutative Hopf algebras.

Deligne's result gives the answer for the case $\mathrm{char}(k)=0$ and nonnegative ranks: comodules over a commutative Hopf algebra.

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  • $\begingroup$ What does "tensor" mean here? Symmetric? Braided? Neither? For symmetric Deligne has a great result. $\endgroup$ Jun 5, 2014 at 17:04

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