Distributive law between monads

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In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other one.

Suppose that (S,\mu^S,\eta^S) and (T,\mu^T,\eta^T) are two monads on a category C. In general, there is no natural monad structure on the composite functor ST. On the other hand, there is a natural monad structure on the functor ST if there is a distributive law of the monad S over the monad T.

Formally, a distributive law of the monad S over the monad T is a natural transformation

l:TS\to ST

such that the diagrams

Distributive law monads mult1.png          Distributive law monads mult2.png
Distributive law monads unit1.png and Distributive law monads unit2.png

commute.

This law induces a composite monad ST with

  • as multiplication: S\mu^T\cdot\mu^STT\cdot SlT,
  • as unit: \eta^ST\cdot\eta^T.

See also[edit]

References[edit]

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