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{{Unreferenced|date=December 2009}}
In [[category theory]], a '''monoidal monad''' <math>(T,\eta,\mu,m)</math> is a [[Monad (category theory)|monad]] <math>(T,\eta,\mu)</math> on a [[monoidal category]] <math>(C,\otimes,I)</math> such that the functor

:<math>T:(C,\otimes,I)\to(C,\otimes,I)</math>

is a [[lax monoidal functor]] with

:<math>m_{A,B}:TA\otimes TB\to T(A\otimes B)</math>

and

:<math>m:I\to TI</math>

as coherence maps, and the natural transformations

:<math>\eta: id \Rightarrow T</math>

and

:<math>\mu:T^2\Rightarrow T</math>

are [[monoidal natural transformation]]s.

By monoidality of <math>\eta</math>, the morphisms <math>m</math> and <math>\eta_I</math> 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.

==Properties==
The [[Monad (category theory)#Algebras for a monad|Kleisli category]] of a monoidal monad has a canonical monoidal structure, induced by the monoidal structure of the monad. The canonical adjunction between <math>C</math> and the Kleisli category is a [[monoidal adjunction]] with respect to this monoidal structure.

{{DEFAULTSORT:Monoidal Monad}}
[[Category:Monoidal categories]]