Invertible Bimodules, Miyashita Action in Monoidal Categories and Azumaya Monoids

TL;DR

This paper develops a unified framework for Miyashita action in monoidal categories, focusing on Azumaya monoids, and demonstrates its applications, including in Hopf algebroids, extending classical results.

Contribution

It introduces a general theory of invertible bimodules and Miyashita action in monoidal categories, generalizing classical results and applying to Azumaya monoids and Hopf algebroids.

Findings

Miyashita action is an isomorphism for certain Azumaya monoids.

The framework unifies previous studies in monoidal categories.

Application to comodules over Hopf algebroids illustrates the theory.

Abstract

In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita's classical result and sheds light on other applications of geometric nature which can not be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.