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A strong monad is a mathematical object defined using category theory that is used in theoretical computer science. In technical terms, 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
This article may be too technical for most readers to understand. (April 2022) |
commute for every object A, B and C (see Definition 3.2 in [1]).
If the monoidal category (C, ⊗, I) is closed then a strong monad is the same thing as a C-enriched monad.
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 .[2]
One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads. More explicitly,
and the conversion between one and the other presentation is bijective.
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