Azumaya Monads and Comonads

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 A by considering a monad (F,m, e) on A endowed with a distributive law λ : FF → FF satisfying the Yang–Baxter equation (BD-law). This allows to introduce an opposite monad (F ,m · λ, e) and a monad structure on FF . The quadruple (F,m, e, λ) is called an Azumaya monad, provided that the canonical comparison functor induces an equivalence between the category A and the category of FF -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 ,⊗, I, τ), for any V-algebra A, the braiding induces a BD-law τA,A : A⊗A→ A⊗A, and A is called left (right) Azumaya, provided the monadA⊗− (resp. −⊗A) is Azumaya. If τ is a symmetry or if the category V admits equalizers and coequalizers, the notions of left and right Azumaya algebras coincide.