Ver_{p^∞} fiber conjecture for symmetric tensor categories

Prove that every symmetric tensor category over an algebraically closed field of positive characteristic p fibers over the Verlinde category Ver_{p^∞}, thereby establishing the positive-characteristic analog of Deligne’s theorem that replaces the role of the supervector spaces category sVec by Ver_{p^∞}.

Background

The paper reviews Deligne’s classification of symmetric tensor categories (STCs) of moderate growth over characteristic zero, where every such STC fibers over the category of supervector spaces sVec. In positive characteristic, Deligne’s theorem fails, and the Verlinde categories Ver_p, Ver_{p^2}, …, Ver_{p^∞} emerge as key incompressible STCs.

Recent work shows that semisimple or Frobenius-exact STCs fiber over certain Verlinde categories, suggesting a broader structural role for Ver_{p^∞}. The stated conjecture proposes Ver_{p^∞} as the universal base over which all STCs in positive characteristic fiber, mirroring the role of sVec in characteristic zero. Confirming this would provide a unified positive-characteristic analog of Deligne’s theorem.