Sometimes people ask, “Does Elm have monads?” And sometimes they ask, “What is a monad? I’ve heard that it is something mysterious and difficult.” This short essay attempts to answer these questions. We begin with an example.

## 1 Chaining Things Together

Elm has functions like these

They are used to chain things together — things of type` Maybe a` or type` Parser a` in the case at hand. As an example, consider the problem of extracting the head of a list. For this we use the function

where

When a list is nonempty,` List.head` returns a value of type` Just a`. In the contrary case, it returns` Nothing`. The example below illustrates this.

Now consider a deeply nested list, say` [[1,2,3]]`. To extract the head of a list, it seems natural to use the function` List.head` twice. Here is a start:

The notation` x |> f` means` f x`. Then` g (f x)` can be written as a "pipeline," namely, as

Notice that` [[1,2,3]] |> List.head` has` Just [1,2,3]` as value. This is a` Maybe` list, and so has the wrong type for a second application of` List.head`. The solution is to use the function

Then successive applications of` List.head` can be chained:

The result,` Just 1`, is a still a value of type` Maybe number`. This is as expected, because we could have computed

Notice how the` Nothing` value is propagated down the pipeline.

# 2 A Pattern

The general pattern for` andThen`, whether it be` Maybe.andThen` or` Parser.andThen`, or something else, is

Here` M`, like` Maybe` or` Parser`, is a type constructor. Or, in Haskell terminology, a functor. Consider the function defined by permuting the arguments of` andThen`:

Up to small differences of notation, this is the bind function in Haskell. It is usually written as an operator, so that

For a type constructor (or functor)` M` to be a monad, it suffices for there to be an associated bind function which satisfies certain properties to be discussed below. Said in the language of Elm, a monad is a type constructor that has an` andThen` function satisfying certain properties.

# 3 Composition

Consider functions` f: a -> b` and` g : b -> c`. One can form the composite function` g o f : a -> c`. More generally, composition can be formed whenever the type of the codomain of` f` is the same as the type of the domain of` g`.

Composition is a good thing. Why? Because composition is what gives us the power to make big things out of smaller things. In programming, that is important because we can reduce complicated things to simpler ones, thereby keeping both the code and ourselves sane.

We cannot compose them as is because the types of the domain and codomain do not match. But observe what we can do with the help of the` andThen` function:

The result is a new composition operator. It can be written in a more suggestive form that “follows the prose:”

Or one can also write it using the bind operator

That is, we use `>>-` to “shove” the value` f x` of type` M b` into the function` g`, which has type` b -> M c`.

Aside. The construction of a new composition law is the point of view of the so-called Kleisli categories. Monadic functions form a category if one uses` o'` instead of` o` for composition.

# 4 Properties

There is a bit more to the monad story than has be recounted so far. The new composition law, like the usual one, should satisfy the identities

and

These identities translate back to identities for` andThen`,` M`, and a function` a -> M a` that in Haskell is called` return`. For` Maybe` in Elm, that is the function that sends a value` x` of type` a` to` Just x`. For` Parser`, it is the function` succeed : a -> Parser a`.

# 5 Summary

• Monads are good because they make it possible to compose functions where this seems to be impossible.
• Monads are not mysterious: they are type constructors (or functors) with certain additional properties, one of which is to make a composition operation possible.

# 6 References

• Kleisli categories, in Bartosz Milewski’s Programming Cafe: Category Theory, Haskell, Concurrency, C++. Studying this reference was what finally turned on the lightbulb for me. My advice is to skip over the C++ parts and concentrate on the Haskell parts. Even better: his book (PDF).
• Joël Quenneville, The Mechanics of Maybe. I’ve adapted the example of deeply nested lists from this article.

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