Opmonoidal monads have been studied under various names. Ieke Moerdijk introduced them as "Hopf monads", while in works of Bruguières and Virelizier they are called "bimonads", by analogy to "bialgebra", reserving the term "Hopf monad" for opmonoidal monads with an antipode, in analogy to "Hopf algebras". An opmonoidal monad is a monad in the 2-category of monoidal categories, oplax monoidal functors and monoidal natural transformations. That means a monad on a monoidal category together with coherence maps and satisfying three axioms that make an opmonoidal functor, and four more axioms that make the unit and the multiplication into opmonoidal natural transformations. Alternatively, an opmonoidal monad is a monad on a monoidal category such that the category of Eilenberg-Moore algebras has a monoidal structure for which the forgetful functor is strong monoidal. An easy example for the monoidal category of vector spaces is the monad, where is a bialgebra. The multiplication and unit of define the multiplication and unit of the monad, while the comultiplication and counit of give rise to the opmonoidal structure. The algebras of this monad are right -modules, which one may tensor in the same way as their underlying vector spaces.
Properties
The Kleisli category of a monoidal monad has a canonical monoidal structure, induced by the monoidal structure of the monad, and such that the free functor is strong monoidal. The canonical adjunction between and the Kleisli category is a monoidal adjunctionwith respect to this monoidal structure, this means that the 2-category has Kleisli objects for monads.
The 2-category of monads in is the 2-category of monoidal monads and it is isomorphic to the 2-category of monoidales in the category of monads, monoidal arrows between them and monoidal cells between them.
The Eilenberg-Moore category of an opmonoidal monad has a canonical monoidal structure such that the forgetful functor is strong monoidal. Thus, the 2-category has Eilenberg-Moore objects for monads.
The 2-category of monads in is the 2-category of monoidal monads and it is isomorphic to the 2-category of monoidales in the category of monads opmonoidal arrows between them and opmonoidal cells between them.
The power set monad. Indeed, there is a function, sending a pair of subsets to the subset. This function is natural in X and Y. Together with the unique function as well as the fact that are monoidal natural transformations, is established as a monoidal monad.
