In category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, a strong monad over a monoidal category

Monoidal category

In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...

  is a monad

Monad (category theory)

In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations...

  together with a natural transformation

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

 , called (tensorial) strength, such that the diagrams

, ,

,

and

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.