Strong monad


In category theory, a strong monad over a monoidal category is a monad together with a natural transformation tA,B : A ⊗ TB → T, called strength, such that the diagrams
commute for every object A, B and C.
If the monoidal category is closed then a strong monad is the same thing as a C-enriched monad.

Commutative strong monads

For every strong monad T on a symmetric monoidal category, a costrength natural transformation can be defined by
A strong monad T is said to be commutative when the diagram
commutes for all objects and.
One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads. More explicitly,
and the conversion between one and the other presentation is bijective.