Strong monad
In category theory, a strong monad is a monad on a monoidal category with an additional natural transformation, called the strength, which governs how the monad interacts with the monoidal product.
Strong monads play an important role in theoretical computer science where they are used to model computation with side effects.
Definition
A strong monad is a monad over a monoidal category together with a natural transformation tA,B : A ⊗ TB → T, called left strength, such that the diagramscommute for every object A, B and C.
Commutative strong monads
For every strong monad T on a symmetric monoidal category, a right strength natural transformation can be defined byA strong monad T is said to be commutative when the diagram
commutes for all objects and.
Properties
The Kleisli category of a commutative monad is symmetric monoidal in a canonical way, see corollary 7 in Guitart and corollary 4.3 in Power & Robison. When a monad is strong but not necessarily commutative, its Kleisli category is a premonoidal category.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