Defining non-degenerate quadratic forms in characteristic 2 symmetric tensor categories

Develop a coherent general definition of non-degenerate quadratic forms for objects in the Verlinde category Ver_4^+ and, more broadly, for symmetric tensor categories over fields of characteristic 2, beyond the ad hoc criterion that a quadratic form q is non-degenerate if and only if its associated symmetric bilinear form β_q is non-degenerate.

Background

In characteristic 2, the usual identification between quadratic forms and symmetric bilinear forms breaks down, and the Frobenius twist creates further complications. The authors discuss standard alternative definitions in the vector-space setting (e.g., non-degeneracy via radicals and reductivity of the associated orthogonal group), but note difficulties in extending these to Ver_4^+ and to arbitrary STCs in characteristic 2.

Because of these obstacles, the paper adopts a simplified working definition (declaring q non-degenerate if β_q is non-degenerate), while explicitly noting the lack of a satisfactory general formulation. A robust definition compatible with categorical structures and the Frobenius functor remains to be formulated.