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,μS,ηS) and (T,μT,ηT) are two monads on a category C. Their composite (as functors) ST is not necessarily a monad! Distributive laws give a way to compose two monads in order to obtain a monad.

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

l:TS\to ST

such that the diagrams

Image:Distributive_law_monads_mult1.png          Image:Distributive_law_monads_mult2.png
Image:Distributive_law_monads_unit1.png and Image:Distributive_law_monads_unit2.png

commute.

This law induces a composite monad ST with

  • as multiplication: SlT.μSμT,
  • as unit: ηSηT.

[edit] See also

[edit] References

  • Jon Beck (1969). "Distributive laws". Lecture Notes in Mathematics 80: 119–140.
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