Question

I’ve noticed that the word “monad” seems to be used in a somewhat inconsistent way. I’ve come to believe that this is because many (if not most) of the monad tutorials out there are written by folks who have only just started to figure monads out themselves (eg: nuclear waste spacesuit burritos), and so the term ends up getting kind of overloaded/corrupted.

In particular, I’m wondering whether the term “monad” can be applied to individual values of types like Maybe, List or IO, or if the term “monad” should really only be applied to the types themselves.

This is a subtle distinction, so perhaps an analogy might make it more clear. In mathematics we have, rings, fields, groups, etc. These terms apply to an entire set of values along with the operations that can be performed on them, rather than to individual elements. For example, integers (along with the operations of addition, negation and multiplication) form a ring. You could say “Integer is a ring”, but you would never say “5 is a ring”.

So, can you say “Just 5 is a monad”, or would that be as wrong as saying “5 is a ring”? I don’t know category theory, but I’m under the impression that it really only makes sense to say “Maybe is a monad” and not “Just 5 is a monad”.

Answer 1

List is a monad, List a is a type, and [] is a List a (an element of a type).

Technically, a monad is a functor with extra structure; and in Haskell we only use functors from the category of Haskell types to itself.

It is thus in particular a “function” which takes a type and returns another type (it has kind * -> *).

List, State s, Maybe, etc are monads. State is not a monad, since it has kind * -> * -> *.

(aside: to confuse matters, Monads are just functors, and if I give myself a partially ordered set A, then it forms a category, with Hom(a, b) = { 1 element } if a <= b and Hom(a, b) = empty otherwise. Now any increasing function f : A -> A forms a functor, and monads are those functions which satisfy x <= f(x) and f(f(x)) <= f(x), hence f(f(x)) = f(x) — monads here are technically “elements of A -> A”. See also closure operators.)

(aside 2: since you appear to know some mathematics, I encourage you to read about category theory. You’ll see among others that algebraic structures can be seen as arising from monads. See this excellent blog entry from the excellent blog by Dan Piponi for a teaser.)