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

## Contents

Opmonoidal monads have been studied under various names; Ieke Moerdijk introduced them as Hopf Monads,[1] 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 ${\displaystyle {\mathsf {OpmonCat}}}$ of monoidal categories, opmonoidal functors and opmonoidal 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.

## Properties

• The category of Kleisli algebras 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 category of Eilenberg-Moore algebras for an opmonoidal monad has a canonical monoidal structure such that the forgetful functor is strong monoidal.[1] This means that 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]

## Examples

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