Distributive law between monads explained

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

(S, \mu ^{S,} η ^{S)} and (T, \mu ^{T,} η ^{T)} 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

l:TS\to ST such that the diagrams

        

         commute.

This law induces a composite monad ST with

• as multiplication:

STST\xrightarrow{SlT}SSTT\xrightarrow{\mu ^{S\mu} ^{T}ST} ,

• as unit:

1 \xrightarrow{ η ^{Sη} ^{T}ST} .

See also

• Distributive property

References

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