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 $(S,\mu ^{S},\eta ^{S})$ 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

commute.

This law induces a composite monad ST with

as multiplication: $STST\xrightarrow{SlT}SSTT\xrightarrow{\mu^S\mu^T}ST$ ,
as unit: $1\xrightarrow{\eta^S\eta^T}ST$ .

References

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Distributive law in nLab
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Distributive law between monads

See also

References