Determinant and rank functions in semisimple pivotal Ab-categories

We investigate and generalize quantum determinants to semisimple spherical and pivotal categories. It is well known that traces are preserved by strong tensor functors; we show on one hand that in fact, weaker conditions on a functor are sufficient to continue preserving traces. On the other hand, we prove that these determinants are well-behaved under strong tensor functors. Further, we introduce a notion of domination rank for objects of a semisimple pivotal category and prove similar properties of the ordinary case. Furthermore, we expand the determinantal and McCoy ranks to introduce a morphism quantum rank function on a semisimple pivotal category.