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. 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

$l:TSto\; ST$

such that the diagrams

         : and commute.

This law induces a composite monad ST with

• as multiplication: $Smu^Tcdotmu^STTcdot\; SlT$,
• as unit: $eta^STcdoteta^T$.

See also

• distributive law

References

• Jon Beck (1969). "Distributive laws". Lecture Notes in Mathematics 80 119–140.
• Michael Barr and Charles Wells Toposes, Triples and Theories. Springer-Verlag.