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. On the other hand, 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

and

commute.

This law induces a composite monad ST with

• as multiplication: $S\mu^T\cdot\mu^STT\cdot SlT$,
• as unit: $\eta^ST\cdot\eta^T$.