Nakayama functor for monads on finite abelian categories

TL;DR

This paper provides explicit formulas for the Nakayama functor in categories of modules over monads on finite abelian categories, with applications to centers of bimodule categories and duals of tensor categories.

Contribution

It introduces explicit formulas for the Nakayama functor in the setting of monads on finite abelian categories, extending understanding in this area.

Findings

Explicit Nakayama functor formulas for monad module categories

Applications to centers of bimodule categories

Formulas for duals of finite tensor categories

Abstract

If $\mathcal{M}$ is a finite abelian category and $\mathbf{T}$ is a linear right exact monad on $\mathcal{M}$, then the category $\mathbf{T}\mbox{-mod}$ of $\mathbf{T}$-modules is a finite abelian category. We give an explicit formula of the Nakayama functor of $\mathbf{T}\mbox{-mod}$ under the assumption that the underlying functor of the monad $\mathbf{T}$ has a double left adjoint and a double right adjoint. As applications, we deduce formulas of the Nakayama functor of the center of a finite bimodule category and the dual of a finite tensor category. Some examples from the Hopf algebra theory are also discussed.