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,^[1] 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.^[2] 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,^[2] 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.^[1]

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).

References

1. Moerdijk, Ieke (23 March 2002). "Monads on tensor categories". Journal of Pure and Applied Algebra 168 (2–3): 189–208. doi:10.1016/S0022-4049(01)00096-2. Retrieved 23 May 2014.
2. Bruguières, Alain; Alexis Virelizier (10 November 2007). "Hopf monads". Advances in Mathematics 215 (2): 679–733. doi:10.1016/j.aim.2007.04.011. Retrieved 23 May 2014.