INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER

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