Supriya Ghosh (Editor)

Strong monad

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Strong monad

In category theory, a strong monad over a monoidal category (C, ⊗, I) is a monad (T, η, μ) together with a natural transformation tA,B : A ⊗ TB → T(A ⊗ B), called (tensorial) strength, such that the diagrams

, , , and

commute for every object A, B and C (see Definition 3.2 in ).

If the monoidal category (C, ⊗, I) 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

t A , B ′ = T ( γ B , A ) ∘ t B , A ∘ γ T A , B : T A ⊗ B → T ( A ⊗ B ) .

A strong monad T is said to be commutative when the diagram

commutes for all objects A and B .

One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads. More explicitly,

  • a commutative strong monad ( T , η , μ , t ) defines a symmetric monoidal monad ( T , η , μ , m ) by
    • m A , B = μ A ⊗ B ∘ T t A , B ′ ∘ t T A , B : T A ⊗ T B → T ( A ⊗ B )
  • and conversely a symmetric monoidal monad ( T , η , μ , m ) defines a commutative strong monad ( T , η , μ , t ) by
    • t A , B = m A , B ∘ ( η A ⊗ 1 T B ) : A ⊗ T B → T ( A ⊗ B )

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

    References

    Strong monad Wikipedia
    (Text) CC BY-SA