Monoidal monad

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In category theory, a monoidal monad (T,η,μ,m) is a monad (T,η,μ) on a monoidal category (C,\otimes,I) such that the functor

T:(C,\otimes,I)\to(C,\otimes,I)

is a lax monoidal functor with

m_{A,B}:TA\otimes TB\to T(A\otimes B)

and

m:I\to TI

as coherence maps, and the natural transformations

\eta:I\Rightarrow T

and

\mu:T^2\Rightarrow T

are monoidal natural transformations.

By monoidality of η, the natural transformations m and ηI are necessarily equal.

[edit] 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.

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