  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.
