Strong Monad - Commutative Strong Monads

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,

• a commutative strong monad defines a symmetric monoidal monad by

• and conversely a symmetric monoidal monad defines a commutative strong monad by

and the conversion between one and the other presentation is bijective.