Distributive law between monads

From Wikipedia, the free encyclopedia

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 $(S,μ S,η S)$ and $(T,μ T,η T)$ are two monads on a category C. In general, there is no natural monad structure on the composite functor ST. A distributive law is exactly what is needed to make ST into a monad.

Formally, a distributive law of the monad S over the monad T is a natural transformation

$l:TS\to ST$

such that the diagrams

Image:Distributive_law_monads_mult1.png

Image:Distributive_law_monads_mult2.png

Image:Distributive_law_monads_unit1.png

and

Image:Distributive_law_monads_unit2.png

commute.

This law induces a composite monad ST with

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

[edit] See also

distributive law

[edit] References

Jon Beck (1969). "Distributive laws". Lecture Notes in Mathematics 80: 119–140.

Michael Barr and Charles Wells (1985). Toposes, Triples and Theories. Springer-Verlag. ISBN 0-387-96115-1.

This category theory-related article is a stub. You can help Wikipedia by expanding it.

Categories: Category theory stubs | Adjoint functors

Views

Interaction

Search