Distributing over the powerset

We saw in the previous post that we can lift the powerset monad to a monad on any category of algebras defined by linear equations. In this short post, we shall tidy some loose ends, leading to another perspective on what we have done.

Eilenberg-Moore Algebras

As we have discussed previously, every finitary monad

arises from the free / forgetful adjunction of an equationally defined category of algebras:

Furthermore, the Eilenberg-Moore category is equivalent to .

Putting these two facts together, if has a presentation only involving linear equations, the previous post shows that the powerset monad lifts to a monad:

Distributive Laws

We have encountered liftings to Eilenberg-Moore categories before. In that case, we were interested in lifting functors to Eilenberg-Moore categories. In this case, we are interested in lifting entire monads.

For monads:

there is a bijection between:

• Liftings of the monad to .
• Distributive laws of type .

This is a topic worthy of further discussion, which we will return to in a later post.

Combining this bijection with the previous observation, if the monad has a presentation by linear equations, there is a distributive law:

Summing Up

Using some standard facts, and Gautam’s results about extending algebraic structure to powersets, we have found sufficient conditions for the existence of a distributive law of over the powerset monad .

Our next steps is to clarify at a higher level of abstraction what is going on, so we can move beyond the powerset to more general monads.