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.