In category theory
, a strong monad
over a monoidal category
is a monad
together with a natural transformation
, called (tensorial
, such that the diagrams
- , ,
- commute for every object , and .
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.