### Eilenberg-Moore

During my attempts to understand the fearsomely difficult Part III course “Introduction to Category Theory” by PT Johnstone, I came across the monadicity of the power-set functor $$\mathbf{Sets} \to \mathbf{Sets}$$. The monad is given by the triple $$(\mathbb{P}, \eta_A: A \to \mathbb{P}(A), \mu_A: \mathbb{PP}(A) \to \mathbb{P}(A))$$, where $$\eta_A: a \mapsto { a }$$, and $$\mu_A$$ is the union operator. So $$\mu_A({ {1, 2 }, {3} }) = {1,2,3 }$$.

It’s easy enough to check that this is a monad. We have a theorem saying that every monad has an associated “Eilenberg-Moore” category - the category of algebras over that monad. What, then, is the E-M category for this monad?

Recall: an algebra over the monad is a pair $$(A, \alpha)$$ where $$A$$ is a set and $$\alpha: \mathbb{P}(A) \to A$$, such that the following two diagrams commute. (That is, $$\alpha$$ here is an operation which takes a collection of elements of $$A$$, and returns an element of $$A$$.)

Aha! The second diagram says that the operation $$\alpha$$ is “massively associative”: however we group up terms and successively apply $$\alpha$$ to them, we’ll come up with the same answer. Mathematica calls this attribute “Flat“ness, when applied to finite sets only.

Moreover, it doesn’t matter what order we feed the elements in to $$\alpha$$, since it works only on sets and not on ordered sets. So $$\alpha$$ is effectively commutative. (Mathematica calls this “Orderless”.)

The first diagram says that $$\alpha$$ applied to a singleton is just the contained element. Mathematica calls this attribute “OneIdentity”.

Finally, $$\alpha(a, a) = \alpha(a)$$, because $$\alpha$$ is implemented by looking at a set of inputs.

So what is an algebra over this monad? It’s a set equipped with an infinitarily-Flat, OneIdentity, commutative operation which ignores repeated arguments. If we forgot that “repeated arguments” requirement, we could use any finite set with any commutative monoid structure; the nonnegative reals with infinity, as a monoid, with addition; and so on. However, this way we’re reduced to monoids which have an operation such that $$a+a = a$$. That’s not many monoids.

What operations do work this way? The Flatten-followed-by-Sort operation in Mathematica obeys this, if the underlying set $$A$$ is a power-set of a well-ordered set. The union operation also works, if the underlying set is a complete poset - so the power-set example is subsumed in that.

Have we by some miracle got every algebra? If we have an arbitrary algebra $$(A, \alpha)$$, we want to define a complete poset which has $$\alpha$$ acting as the union. So we need some ordering on $$A$$; and if $$x \leq y$$, we need $$\alpha({x, y}) = y$$. That looks like a fair enough definition to me. It turns out that this definition just works.

So the Eilenberg-Moore category of the covariant power-set functor is just the category of complete posets.

(Subsequently, I looked up the definition of “complete poset”, and it turns out I mean “complete lattice”. I’ve already identified the need for unions of all sets to exist, so this is just a terminology issue. A complete poset only has sups of directed sequences. A complete lattice has all sups.)