Towards higher Frobenius functors for symmetric tensor categories

Abstract We develop theory and examples of monoidal functors on tensor categories in positive characteristic that generalise the Frobenius functor from [5, 17, 22]. 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 $$\operatorname {Ver}_{p^n}$$ Ver p n in terms of representation categories of elementary abelian p -groups. This leads to families of functors relating to $$\operatorname {Ver}_{p^n}$$ Ver p n that we conjecture, and partially show, to exhibit the characteristic properties of the Frobenius functor relating to $$\operatorname {Ver}_p$$ Ver p . In particular, we conjecture some of these functors to detect categories that fibre over $$\operatorname {Ver}_{p^n}$$ Ver p n .