Azumaya monads and comonads

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.