Internal natural transformations and Frobenius algebras in the Drinfeld center
Abstract
For M and N finite module categories over a finite tensor category C, the category RexC(M,N) of right exact module functors is a finite module category over the Drinfeld center Z(C). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map C-C-bimodule functors to objects of Z(C), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if M and N are exact C-modules and C is pivotal, then the Z(C)-module RexC(M,N) is exact. We compute its relative Serre functor and show that if M and N are even pivotal module categories, then RexC(M,N) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in Z(C).