Internal natural transformations and Frobenius algebras in the Drinfeld center

TL;DR

This paper investigates internal natural transformations and Frobenius algebras within the Drinfeld center, revealing their structure and properties in the context of finite tensor categories and module categories.

Contribution

It introduces the concept of internal natural transformations, expresses them via ends, and explores their relation to Frobenius algebras in the Drinfeld center, extending understanding of module categories.

Findings

Rex_C(M,N) is exact when M and N are exact C-modules and C is pivotal.

Computed the relative Serre functor for Rex_C(M,N).

Internal Ends in Rex_C(M,N) form Frobenius algebras in Z(C).

Abstract

For M and N finite module categories over a finite tensor category C, the category Rex_C(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 Rex_C(M,N) is exact. We compute its relative Serre functor and show that if M and N are even pivotal module categories, then Rex_C(M,N) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in Z(C).