Monoidal monad

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

All of the above can be compressed into the statement that a monoidal monad is a monad in the 2-category ${\displaystyle {\mathsf {MonCat}}}$ of monoidal categories, lax monoidal functors, and monoidal natural transformations.

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 ${\displaystyle (T,\eta ,\mu )}$ in the 2-category of ${\displaystyle {\mathsf {OpMonCat}}}$ monoidal categories, oplax monoidal functors and monoidal natural transformations. That means a monad ${\displaystyle (T,\eta ,\mu )}$ on a monoidal category ${\displaystyle (C,\otimes ,I)}$ together with coherence maps ${\displaystyle T^{A,B}:T(A\otimes B)\to TA\otimes TB}$ and ${\displaystyle T^{0}:TI\to I}$ satisfying three axioms that make an opmonoidal functor, and four more axioms that make the unit ${\displaystyle \eta }$ and the multiplication ${\displaystyle \mu }$ 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 ${\displaystyle \operatorname {Vect} }$ of vector spaces is the monad ${\displaystyle -\otimes A}$, where ${\displaystyle A}$ is a bialgebra.^[2] The multiplication and unit of ${\displaystyle A}$ define the multiplication and unit of the monad, while the comultiplication and counit of ${\displaystyle A}$ give rise to the opmonoidal structure. The algebras of this monad are right ${\displaystyle A}$-modules, which one may tensor in the same way as their underlying vector spaces.

• 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 ${\displaystyle C}$ and the Kleisli category is a monoidal adjunction with respect to this monoidal structure, this means that the 2-category ${\displaystyle {\mathsf {MonCat}}}$ has Kleisli objects for monads.
• The 2-category of monads in ${\displaystyle {\mathsf {MonCat}}}$ is the 2-category of monoidal monads ${\displaystyle {\mathsf {Mnd(MonCat)}}}$ and it is isomorphic to the 2-category ${\displaystyle {\mathsf {Mon(Mnd(Cat))}}}$ of monoidales (or pseudomonoids) in the category of monads ${\displaystyle {\mathsf {Mnd(Cat)}}}$, (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 ${\displaystyle {\mathsf {OpmonCat}}}$ has Eilenberg-Moore objects for monads.^[3]
• The 2-category of monads in ${\displaystyle {\mathsf {OpmonCat}}}$ is the 2-category of monoidal monads ${\displaystyle {\mathsf {Mnd(OpmonCat)}}}$ and it is isomorphic to the 2-category ${\displaystyle {\mathsf {Opmon(Mnd(Cat))}}}$ of monoidales (or pseudomonoids) in the category of monads ${\displaystyle {\mathsf {Mnd(Cat)}}}$ opmonoidal arrows between them and opmonoidal cells between them.^[4]

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

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

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