The relative modular object and Frobenius extensions of finite Hopf algebras

For a certain kind of tensor functor F:C→DF:C→D, we define the relative modular object χF∈DχF∈D as the “difference” between a left adjoint and a right adjoint of F  . Our main result claims that, if CC and DD are finite tensor categories, then χFχF can be written in terms of a categorical analogue of the modular function on a Hopf algebra. Applying this result to the restriction functor associated to an extension A/BA/B of finite-dimensional Hopf algebras, we recover the result of Fischman, Montgomery and Schneider on the Frobenius type property of A/BA/B. We also apply our results to obtain a “braided” version and a “bosonization” version of the result of Fischman et al.