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.

Suppose that $(S,{\mu }^S,{\eta }^S)$ and $(T,{\mu }^T,{\eta }^T)$ 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

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: $STST\stackrel{SlT}{\to }SSTT\stackrel{{\mu }^S{\mu }^T}{\to }ST$,
• as unit: $1\stackrel{{\eta }^S{\eta }^T}{\to }ST$.

See also

• Distributive property

References

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