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We develop theory and examples of monoidal functors on tensor categories in positive characteristic that generalise the Frobenius functor from Os, EOf, Tann. The latter has proved to be a powerful tool in the ongoing classification of tensor categories of moderate growth, and we demonstrate the similar potential of the generalisations. More explicitly, we describe a new construction of the generalised Verlinde categories Ver䂞 in terms of representation categories of elementary abelian p-groups. This leads to families of functors relating to Ver䂞 that we conjecture, and partially show, to exhibit the characteristic properties of the Frobenius functor relating to Verₚ.
Coulembier et al. (Wed,) studied this question.