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,μ^S,η^S) and (T,μ^T,η^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

such that the diagrams

        

and

commute.

This law induces a composite monad ST with

• as multiplication: ,
• as unit: .

[edit] See also

• distributive law

[edit] References

• Jon Beck (1969). "Distributive laws". Lecture Notes in Mathematics 80: 119–140. doi:10.1007/BFb0083084. 

• Michael Barr and Charles Wells (1985). Toposes, Triples and Theories. Springer-Verlag. ISBN 0-387-96115-1.