category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A spherical category is a monoidal category with duals that behaves as if its morphisms can be drawn and moved around on a sphere.
A spherical category is a pivotal category where the left and right trace operations coincide on all objects.
spherical category
The definition is originally due to
A review is in section 2.3 of
More is in