Monoidal monad

In category theory, a monoidal monad $(T,\eta ,\mu ,T_{A,B},T_{0})$ 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 $\eta$ and $\mu$ are monoidal natural transformations. In other words, $T$ is equipped with coherence maps $T_{A,B}:TA\otimes TB\to T(A\otimes B)$ and $T_{0}:I\to TI$ satisfying certain properties (again: they are lax monoidal), and the unit $\eta :id\Rightarrow T$ and multiplication $\mu :T^{2}\Rightarrow T$ are monoidal natural transformations. By monoidality of $\eta$ , the morphisms $T_{0}$ and $\eta _{I}$ are necessarily equal.

All of the above can be compressed into the statement that a monoidal monad is a monad in the 2-category ${\mathsf {MonCat}}$ 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",[1] while in works of Bruguières and Virelizier they are called "bimonads", by analogy to "bialgebra",[2] reserving the term "Hopf monad" for opmonoidal monads with an antipode, in analogy to "Hopf algebras".

An opmonoidal monad is a monad $(T,\eta ,\mu )$ in the 2-category of ${\mathsf {OpMonCat}}$ monoidal categories, oplax monoidal functors and monoidal natural transformations. That means a monad $(T,\eta ,\mu )$ on a monoidal category $(C,\otimes ,I)$ together with coherence maps $T^{A,B}:T(A\otimes B)\to TA\otimes TB$ and $T^{0}:TI\to I$ satisfying three axioms that make an opmonoidal functor, and four more axioms that make the unit $\eta$ and the multiplication $\mu$ 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.[1][3]

An easy example for the monoidal category $\operatorname {Vect}$ of vector spaces is the monad $-\otimes A$ , where $A$ is a bialgebra.[2] The multiplication and unit of $A$ define the multiplication and unit of the monad, while the comultiplication and counit of $A$ give rise to the opmonoidal structure. The algebras of this monad are right $A$ -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 $C$ and the Kleisli category is a monoidal adjunction with respect to this monoidal structure, this means that the 2-category ${\mathsf {MonCat}}$ has Kleisli objects for monads.
The 2-category of monads in ${\mathsf {MonCat}}$ is the 2-category of monoidal monads ${\mathsf {Mnd(MonCat)}}$ and it is isomorphic to the 2-category ${\mathsf {Mon(Mnd(Cat))}}$ of monoidales (or pseudomonoids) in the category of monads ${\mathsf {Mnd(Cat)}}$ , (lax) monoidal arrows between them and monoidal cells between them.[4]
The Eilenberg-Moore category of an opmonoidal monad has a canonical monoidal structure such that the forgetful functor is strong monoidal.[1] Thus, the 2-category ${\mathsf {OpmonCat}}$ has Eilenberg-Moore objects for monads.[3]
The 2-category of monads in ${\mathsf {OpmonCat}}$ is the 2-category of monoidal monads ${\mathsf {Mnd(OpmonCat)}}$ and it is isomorphic to the 2-category ${\mathsf {Opmon(Mnd(Cat))}}$ of monoidales (or pseudomonoids) in the category of monads ${\mathsf {Mnd(Cat)}}$ opmonoidal arrows between them and opmonoidal cells between them.[4]

Examples

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

The power set monad $(\mathbb {P} ,\varnothing ,\cup )$ . Indeed, there is a function $\mathbb {P} (X)\times \mathbb {P} (Y)\to \mathbb {P} (X\times Y)$ , sending a pair $(X'\subseteq X,Y'\subseteq Y)$ of subsets to the subset $\{(x,y)\mid x\in X'{\text{ and }}y\in Y'\}\subseteq X\times Y$ . This function is natural in X and Y. Together with the unique function $\{1\}\to \mathbb {P} (\varnothing )$ as well as the fact that $\mu ,\eta$ are monoidal natural transformations, $(\mathbb {P}$ is established as a monoidal 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

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.
Bruguières, Alain; Alexis Virelizier (2007). "Hopf monads". Advances in Mathematics. 215 (2): 679–733. doi:10.1016/j.aim.2007.04.011.
McCrudden, Paddy (2002). "Opmonoidal monads". Theory and Applications of Categories. 10 (19): 469–485. CiteSeerX 10.1.1.13.4385.
Zawadowski, Marek (2011). "The Formal Theory of Monoidal Monads The Kleisli and Eilenberg-Moore objects". Journal of Pure and Applied Algebra. 216 (8–9): 1932–1942. arXiv:1012.0547. doi:10.1016/j.jpaa.2012.02.030. S2CID 119301321.

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[moerdijk-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.

[bruguieres-2] Bruguières, Alain; Alexis Virelizier (2007). "Hopf monads". Advances in Mathematics. 215 (2): 679–733. doi:10.1016/j.aim.2007.04.011.

[:0-3] McCrudden, Paddy (2002). "Opmonoidal monads". Theory and Applications of Categories. 10 (19): 469–485. CiteSeerX 10.1.1.13.4385.

[:1-4] Zawadowski, Marek (2011). "The Formal Theory of Monoidal Monads The Kleisli and Eilenberg-Moore objects". Journal of Pure and Applied Algebra. 216 (8–9): 1932–1942. arXiv:1012.0547. doi:10.1016/j.jpaa.2012.02.030. S2CID 119301321.