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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.

Suppose that and are two monads on a category C. In general, there is no natural monad structure on the composite functor ST. However, 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

such that the diagrams

Distributive law monads mult1          Distributive law monads unit1
Distributive law monads mult2          Distributive law monads unit2

commute.

This law induces a composite monad ST with

  • as multiplication: ,
  • as unit: .

Examples

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See also

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References

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  • Beck, Jon (1969). "Distributive laws". Seminar on Triples and Categorical Homology Theory, ETH 1966/67. Lecture Notes in Mathematics. Vol. 80. pp. 119–140. doi:10.1007/BFb0083084. ISBN 978-3-540-04601-1.