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.