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

${\displaystyle l:TS\to ST}$

such that the diagrams

commute.

This law induces a composite monad ST with

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