Azumaya monads and comonads (Q2352961)

Azumaya algebras \(\mathcal{A} = (A, m, e)\) over a commutative ring \(R\) are characterized by the fact that the functor \(A \otimes_R -\) induces an equivalence between the category of \(R\)-modules and the category of \((A, A)\)-bimodules. In this paper, the authors 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-law) (see definition 2.2 in [\textit{S. Kasangian} et al., Theory Appl. Categ. 13, 129--146 (2004; Zbl 1085.18005)]). Some properties and characterizations of Azumaya monads are given. Also the authors defined Azumaya comonads as a dual notion of Azumaya monads and investigates the relationship between the Azumaya properties of monad \(\mathcal{F}\) and the comonads \(\mathcal{R}\). Further, the results about Azumaya comonads provides an extensive theory of Azumaya coalgebras in braided categories \(\mathcal{V}\), and the basics for this are described in last section.