Azumaya monads and comonads

TL;DR

This paper generalizes the concept of Azumaya algebras to monads and comonads in arbitrary categories, exploring their properties, dualities, and applications in braided monoidal categories and coalgebra theory.

Contribution

It introduces Azumaya monads and comonads, extending Azumaya algebra concepts to categorical frameworks with new characterizations and dualities.

Findings

Defined Azumaya monads via distributive laws satisfying the Yang-Baxter equation.

Established conditions for Azumaya monads and comonads, especially with right adjoints.

Characterized Azumaya coalgebras in terms of dual Azumaya algebras over rings.

Abstract

The definition of Azumaya algebras over commutative rings $R$ require the tensor product of modules over $R$ and the twist map for the tensor product of any two $R$-modules. Similar constructions are available in braided monoidal categories and Azumaya algebras were defined in these settings. Here we introduce Azumaya monads on any category $\A$ by considering a monad $\bF$ on $\A$ endowed with a distributive law $\lambda: FF\to FF$ satisfying the Yang-Baxter equation (BD-law). This allows to introduce an {\em opposite monad} $\bF^\la$ and a monad structure on $FF^\la$. For an {\em Azumaya monad} we impose the condition that the canonical comparison functor induces an equivalence between the category $\A$ and the category of $\bF\bF^\la$-modules. Properties and characterisations of these monads are studied, in particular for the case when $F$ allows for a right adjoint functor. Dual to Azumaya monads we define {\em Azumaya comonads} and investigate the interplay between these notions. In braided categories $(\V,\ot,I,\tau)$, for any $\V$-algebra $A$, the braiding induces a BD-law $\tau_{A,A}:A\ot A\to A\ot A$ and $A$ is called left (right) Azumaya, provided the monad $A\ot-$ (resp. $-\ot A$) is Azumaya. If $\tau$ is a symmetry, or if the category $\V$ admits equalisers and coequalisers, the notions of left and right Azumaya algebras coincide. The general theory provides the definition of coalgebras in $\V$. Given a cocommutative $\V$-coalgebra $\bD$, coalgebras $\bC$ over $\bD$ are defined as coalgebras in the monoidal category of $\bD$-comodules and we describe when these have the Azumaya property. In particular, over commutative rings $R$, a coalgebra $C$ is Azumaya if and only if the dual $R$-algebra $C^*=\Hom_R(C,R)$ is an Azumaya algebra.