N-spherical Functors and Tensor Categories

We apply the recently introduced notion, due to Dyckerhoff, Kapranov, and Schechtman, of N-spherical functors of stable infinity categories, which generalise spherical functors, to the setting of monoidal categories. We call an object N-bounded if the corresponding regular endofunctor on the derived category is N-spherical. Besides giving new examples of N-spherical functors, the notion of N-bounded objects gives surprising connections with Jones-Wenzl idempotents, Frobenius-Perron dimensions, and central conjectures in the field of symmetric tensor categories in positive characteristic.