Azumaya Monads and Comonads

Abstract

The definition of Azumaya algebras over commutative rings $R$ requires 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 $\mathbb{A}$ by considering a monad $(F,m,e)$ on $\mathbb{A}$ endowed with a distributive law $\lambda: FF\to FF$ satisfying the Yang–Baxter equation (BD%please define -law). This allows to introduce an opposite monad $(F^\lambda,m\cdot \lambda,e)$ and a monad structure on $FF^\lambda$. The quadruple $(F,m,e,\lambda)$ is called an Azumaya monad, provided that the canonical comparison functor induces an equivalence between the category $\mathbb{A}$ and the category of $FF^\lambda$-modules. Properties and characterizations of these monads are studied, in particular for the case when $F$ allows for a right adjoint functor. Dual to Azumaya monads, we define Azumaya comonads and investigate the interplay between these notions. In braided categories (V$,\otimes,I,\tau)$, for any V-algebra $A$, the braiding induces a BD-law $\tau_{A,A}:A\otimes A\to A\otimes A$, and $A$ is called left (right) Azumaya, provided the monad $A\otimes-$ (resp. $-\otimes A$) is Azumaya. If $\tau$ is a symmetry or if the category V admits equalizers and coequalizers, the notions of left and right Azumaya algebras coincide