INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER

Abstract

For ℳ and mathcal{N} finite module categories over a finite tensor category mathcal{C} , the category mathrm{mathcal{R}}{ex}_{mathcal{C}} (ℳ, mathcal{N} ) of right exact module functors is a finite module category over the Drinfeld center mathcal{Z} ( mathcal{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 mathcal{C} - mathcal{C} -bimodule functors to objects of mathcal{Z} ( mathcal{C} ), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and mathcal{N} are exact mathcal{C} -modules and mathcal{C} is pivotal, then the mathcal{Z} ( mathcal{C} )-module mathrm{mathcal{R}}{ex}_{mathcal{C}} (ℳ, mathcal{N} ) is exact. We compute its relative Serre functor and show that if ℳ and mathcal{N} are even pivotal module categories, then mathrm{mathcal{R}}{ex}_{mathcal{C}} (ℳ, mathcal{N} ) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in mathcal{Z} ( mathcal{C} ).