In category theory, a branch of mathematics, a monoidal monad is a monad on a monoidal category such that the functor is a lax monoidal functor and the natural transformations are monoidal natural transformations satisfying certain properties are monoidal natural transformations are necessarily equal.

All of the above can be compressed into the statement that a monoidal monad is a monad in the 2-category of monoidal categories, lax monoidal functors, and monoidal natural transformations.

Opmonoidal monads

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 in on 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.

is a bialgebra.-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 and the Kleisli category is a monoidal adjunction has Kleisli objects for monads.

• , (lax) 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. has Eilenberg-Moore objects for monads.

• opmonoidal arrows between them and opmonoidal cells between them.

Examples

The following monads on the category of sets, with its cartesian monoidal structure, are monoidal monads:

• The power set. This function is natural in X and Y is established as a monoidal monad.

• The probability distribution (Giry) monad.

The following monads on the category of sets, with its cartesian monoidal structure, are not monoidal monads

• is commutative).

References