In category theory, an abstract branch of mathematics, distributive laws between monadss are a way to express abstractly that two algebraic structures distribute one over the other.

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

such that the diagrams

commute.

This law induces a composite monad ST with

• ,

• .

Examples

Monoids

Informally, one might say that the free monoid on a set is given by "the free semigroup, plus an identity element". We can formalise this intuition, using the fact that both the free semigroup and the "free element" functors are monads. Explicitly, consider the monads

1. ,

(here addition denotes disjoint union), with unit maps given by:

1. .

.

as the empty word; explicitly:

can be incorporated into the semigroup as a unit element (thereby making the structure into a monoid). Let us check that all the diagrams in the definition commute.

1. ,

.

1. ,

, as required.

1. ,

.

1. .

In summary, the four commutation requirements encode the following ideas:

1. .

2. truly is a disjoint union.

3. .

4. as the empty word.

Rings

This is the original example given by Beck, to (the underlying set of) the free commutative group and the free monoid on it, respectively. We shall regard the formal group give the free ring to show that it's also a monad.

, the multiplicative monoid - contrary to the usual convention; see the next section.

Terminology

a distributive law of S over T. is instead said to be a distributive law of T over S; this is because, arguably, the latter convention corresponds more closely to the usual distributive law between multiplication and addition - multiplication is typically said to distribute over addition, by which one means that a product of sums can be rewritten as a sum of products (see Examples). Nevertheless, both terminologies are in use in different sources, and the original is what this article uses.

Related notions

BD-law

is called a BD-law if:

1. (D) is a distributive law, and

2. (B) it satisfies the Yang-Baxter equation:

   1. ).

A BD-law is a BCD-law-commutative", meaning that

1. (C)

   1. ).

Condition (B) is called a Yang-Baxter equation because it literally is the Yang-Baxter equation defined on monoid objects is viewed as a monoid in the category of endofunctors.

Entwining

an algebra a coalgebra over a shared field, an entwining, induced by the counit.

See also

• Distributive property

References