We can think of a monad on Set Set as describing some sort of algebraic gadget equipped with a bunch of operations obeying a bunch of equations. This works very nicely if we restrict attention to finitary monads, which correspond to Lawvere theories: then the operations I’m talking about are all ‘finitary’, taking some finite set of inputs. But we can also generalize to higher cardinalities: for any cardinal α, a monad of rank α on Set Set describes some sort of gadget with operations of arity at most α.

There are nastier gadgets that have no upper bound on the arity of their operations, like complete semilattices, also known as suplattices. The point is that in such a thing, any subset has a least upper bound, no matter how large its cardinality. These are algebras of a ‘monad without rank’ — which makes me think of someone in the army who is not a private, not a lieutenant, not a colonel, not a general….

Anyway, this viewpoint on monads helps me get a feeling for commutative monads: these describe algebraic gadgets with a bunch of operations that all commute with each other, and perhaps obey other equations as well.

I want to ask some questions about a categorified version of this story, involving pseudocommutative 2-monads.