Monoidal monad

In category theory, a monoidal monad $(T,\eta,\mu,m)$ is a monad $(T,\eta,\mu)$ on a monoidal category $(C,\otimes,I)$ such that the functor

$T:(C,\otimes,I)\to(C,\otimes,I)$

is a lax monoidal functor and the natural transformations $\eta,\mu$ are monoidal natural transformations. In other words, $T$ is equipped with coherence maps

$m_{A,B}:TA\otimes TB\to T(A\otimes B)$

and

$m:I\to TI$

satisfying certain properties, and its structure maps

$\eta: id \Rightarrow T$

and

$\mu:T^2\Rightarrow T$

must be monoidal with respect to $(C,\otimes,I)$. By monoidality of $\eta$, the morphisms $m$ and $\eta_I$ are necessarily equal.

This is equivalent to saying that a monoidal monad is a monad in the 2-category MonCat of monoidal categories, monoidal functors, and monoidal natural transformations.

Hopf monads and bimonads

Ieke Moerdijk introduced the notion of a Hopf monad, which is an opmonoidal monad, that is, a monad with coherence morphisms $m^{A,B}:T(A\otimes B) \to TA\otimes TB$ and $m^0:TI\to I$ and opmonoidal natural transformations as multiplication and left/right units.

An easy example for the category $\operatorname{Vect}$ of vector spaces is the monad $- \otimes A$, where $A$ is a bialgebra. The multiplication in $A$ then defines the multiplication of the monad, while the comultiplication gives rise to the opmonoidal structure. The algebras of this monad are just right $A$-modules.

In works of Bruguières and Virelizier, this concept has been renamed bimonad, by analogy to "bialgebra". They reserve the term "Hopf monad" for bimonads with an antipode, in analogy to "Hopf algebras".

Properties

• The Kleisli category of a monoidal monad has a canonical monoidal structure, induced by the monoidal structure of the monad. The canonical adjunction between $C$ and the Kleisli category is a monoidal adjunction with respect to this monoidal structure.
• The Eilenberg-Moore category (the category of algebras) of a Hopf monad (in Moerdijk's nomenclature) has a canonical monoidal structure.

Examples

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

• The power set monad.
• The probability distributions (Giry) monad.

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

• If $M$ is a monoid, then $X\mapsto X\times M$ is a monad, but in general there is no reason to expect a monoidal structure on it (unless $M$ is commutative).