Distributive law between monads

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 [math]\displaystyle{ (S,\mu^S,\eta^S) }[/math] and [math]\displaystyle{ (T,\mu^T,\eta^T) }[/math] 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

[math]\displaystyle{ l:TS\to ST }[/math]

such that the diagrams

        

        

commute.

This law induces a composite monad ST with

• as multiplication: [math]\displaystyle{ STST\xrightarrow{SlT}SSTT\xrightarrow{\mu^S\mu^T}ST }[/math],
• as unit: [math]\displaystyle{ 1\xrightarrow{\eta^S\eta^T}ST }[/math].

See also

• distributive law

References

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