# Functors, Applicatives, And Monads In Pictures (In ReasonML)

This is a translation of Functors, Applicatives, And Monads In Pictures from Haskell into ReasonML.

I take no credit for this work, and if you enjoy this post be sure to say thanks to Aditya Bhargava (@_egonschiele) the author of the original version.

Here's a simple value: And we know how to apply a function to this value: Simple enough. Let's extend this by saying that any value can be in a context. For now you can think of a context as a box that you can put a value in: Now when you apply a function to this value, you'll get different results depending on the context. This is the idea that Functors, Applicatives, Monads, Arrows etc are all based on.

Let's create a `Maybe` data type that defines two related contexts:

``````module Maybe = {
type t('a) =
| Nothing
| Just('a);
}
``````

In a second we'll see how function application is different when something is a `Just(a)` versus a `Nothing`. First let's talk about Functors!

## Functors

When a value is wrapped in a context, you can't apply a normal function to it: This is where `fmap` comes in. `fmap` is from the street, `fmap` is hip to contexts. `fmap` knows how to apply functions to values that are wrapped in a context.

Suppose you want to apply `(+3)` to `Just(2)`. We can implement `fmap`:

``````Maybe.fmap((+)(3), Just(2));
// Just(5)
`````` Bam! `fmap` shows us how it's done!

## Just what is a Functor, really?

`Functor` is a class of functions that define an `fmap` function.

In Haskell they are defined as a type class.

In ReasonML we don't have type classes yet, but they are being worked on in OCaml, and therefore ReasonML.

Here's the definition: A `Functor` is any data type that defines how `fmap` applies to it. Here's how `fmap` works: So we can do this:

``````Maybe.fmap((+)(3), Just(2));
// Just(5)
``````

This specifies how `fmap` applies to `Just`s and `Nothing`s:

Let's add `fmap` to our `Maybe` module:

``````let fmap = (f, m) => {
switch (m) {
| Nothing => Nothing
| Just(a) => Just(f(a))
};
};
``````

Here's what is happening behind the scenes when we write `Maybe.fmap((+)(3), Just(2));`: So then you're like, alright `fmap`, please apply `(+3)` to a `Nothing`?

``````Maybe.fmap((+)(3), Nothing)
// Nothing
``````

Like Morpheus in the Matrix, `fmap` knows just what to do; you start with `Nothing`, and you end up with `Nothing`! `fmap` is zen. Now it makes sense why the `Maybe` data type exists. For example, here's how you work with a database record in a language without `Maybe`:

``````post = Post.find_by_id(1)
if post
return post.title
else
return nil
end
``````

Let's create a simple `Post` module in ReasonML with these functions.

``````module Post = {
type t = {
id: int,
title: "string,"
};
let make = (id, title) => {id, title};
let fmap = (f, post) => f(post);
let getPostTitle = post => post.title;
let findPost = id => make(id, "Post #" ++ string_of_int(id));
};
``````

Now we can write:

``````Post.(fmap(getPostTitle, findPost(1)));
``````

If `findPost` returns a post, we will get the title with `getPostTitle`. If it returns `Nothing`, we will return `Nothing`! Pretty neat, huh?

In Haskell, `<\$>` is the common infix version of `fmap`.

In ReasonML we can create an equivalent alias. Adding to our `Post` module:

``````let (<\$>) = fmap;
``````

So we can now write:

``````Post.(getPostTitle <\$> findPost(1));
``````

Here's another example: what happens when you apply a function to a list? Lists can operate as functors too!

In ReasonML we can make use of the list map function:

``````List.map
``````

Okay, okay, one last example: what happens when you apply a function to another function?

For this case we can define a `Function` module that has `fmap`:

``````module Function = {
let fmap = (f, g, x) => f(g(x));
};
``````

So we can now write:

``````Function.fmap((+)(3), (+)(1));
``````

Here's a function: Here's a function applied to another function: The result is just another function!

``````let foo = Function.fmap((+)(3), (+)(2));
foo(10);
// 15
``````

So functions can be Functors too!

When you use `fmap` on a function, you're just doing function composition!

## Applicatives

Applicatives take it to the next level. With an applicative, our values are wrapped in a context, just like Functors: But our functions are wrapped in a context too! Yeah. Let that sink in. Applicatives don't kid around. They know how to apply a function wrapped in a context to a value wrapped in a context:

Applicatives define an `apply` function, also written as `<*>` in Haskell, which we can create an alias for in ReasonML. Let's add the applicative functions to our `Maybe` module:

``````let apply = (mf, mv) => {
switch (mv) {
| Nothing => Nothing
| Just(v) =>
switch (mf) {
| Nothing => Nothing
| Just(f) => Just(f(v))
}
};
};

let (<*>) = apply;
``````

Here's an example using them:

``````Maybe.(Just((+)(3)) <*> Just(2));
// Just(5)
``````

Let's also define applicative functions for a list. We'll create a `MyList` module to avoid name clashes with the built in `List` module:

``````module MyList = {
type apply('a, 'b) = (list('a => 'b), list('a)) => list('b);

let apply: apply('a, 'b) =
(fs, xs) => List.flatten(List.map(f => List.map(f, xs), fs));

let (<*>) = apply;
};
``````

Using `<*>` can lead to some interesting situations. For example:

``````let funList = [(*)(2), (+)(3)];
let valList = [1, 2, 3];
MyList.(funList <*> valList);
// [2, 4, 6, 4, 5, 6]
`````` Here's something you can do with Applicatives that you can't do with Functors. How do you apply a function that takes two arguments to two wrapped values?

``````Maybe.((+) <\$> Just(5));
// Just((+)(5))

Maybe.(Just((+)(5)) <\$> Just(4));
// ERROR ??? WHAT DOES THIS EVEN MEAN WHY IS THE FUNCTION WRAPPED IN A JUST
``````

Applicatives:

``````Maybe.((+) <\$> Just(5));
// Just((+)(5))

Maybe.(Just((+)(5)) <*> Just(3));
// Just(8)
``````

`Applicative` pushes `Functor` aside. "Big boys can use functions with any number of arguments," it says. "Armed with `<\$>` and `<*>`, I can take any function that expects any number of unwrapped values. Then I pass it all wrapped values, and I get a wrapped value out! AHAHAHAHAH!"

``````Maybe.((*) <\$> Just(5) <*> Just(3));
``````

1. Get a PhD in computer science.
2. Throw it away because you don't need it for this section!

Functors apply a function to a wrapped value: Applicatives apply a wrapped function to a wrapped value: Monads apply a function that returns a wrapped value to a wrapped value.

Monads have a function `bind`, or the operator alias `>>=` to do this.

Let's see an example.

First, we'll need to add bind to our good ol' `Maybe` module:

``````let bind = (mv, f) => {
switch (mv) {
| Nothing => Nothing
| Just(v) => f(v)
};
};

let (>>=) = bind;
`````` Suppose `half` is a function that only works on even numbers:

Let's write `half` in ReasonML (we'll also need to define `even` and `odd` functions):

``````/*
Mutually recursive function
https://ocaml.org/learn/tutorials/labels.html
*/
let rec even = x =>
if (x <= 0) {
true;
} else {
odd(x - 1);
}
and odd = x =>
if (x <= 0) {
false;
} else {
even(x - 1);
};

let half = x =>
if (even(x)) {
Maybe.Just(x / 2);
} else {
Nothing;
};
`````` What if we feed it a wrapped value? We need to use `>>=` to shove our wrapped value into the function. Here's a photo of `>>=`: Here's how it works:

``````Maybe.(Just(3) >>= half);
// Nothing

Maybe.(Just(4) >>= half);
// Just(2)

Maybe.(Nothing >>= half);
// Nothing
``````

What's happening inside? `Monad` defines a `bind` (or `>>=`) function: Let's make our `Maybe` into a monad by adding the `bind` functions.

``````let bind = (mv, f) => {
switch (mv) {
| Nothing => Nothing
| Just(v) => f(v)
};
};

let (>>=) = bind;
``````

Here it is in action with a `Just(3)`! And if you pass in a `Nothing` it's even simpler: You can also chain these calls:

``````Maybe.(Just(20) >>= half >>= half >>= half);
``````  Cool stuff! So now we have implemented `Maybe` to be a `Functor`, an `Applicative`, and a `Monad`.

Now let's mosey on over to another example and create an `IO` monad: The `IO` monad exists in Haskell, but we will declare our own in ReasonML.

Specifically three functions. `getLine` takes no arguments and gets user input: `readFile` takes a string (a filename) and returns that file's contents: `putStrLn` takes a string and prints it: Our `IO` module might look something like this (leaving out implementation details of the helper functions):

``````module IO = {
type t = Js.Promise.t(string);

type bind('a, 'b) = (t, string => t) => t;
let bind: bind('a, 'b) = (pa, f) => pa |> Js.Promise.then_(a => f(a));

let (>>=) = bind;

type getLine = unit => t;
let getLine = ...

type readFile = string => t;

type putStrLn = string => t;
let putStrLn = ...
};
``````

If you're interested, the full source code is available.

All three functions take a regular value (or no value) and return a wrapped `Promise` value. We can chain all of these using `>>=`!

``````IO.(getLine() >>= readFile >>= putStrLn);
``````

Aw yeah! Front row seats to the monad show!

## Conclusion

1. A functor is a data type that implements the `fmap` function.
2. An applicative is a data type that implements the `apply` function.
3. A monad is a data type that implements the `bind` function.

The `Maybe` module in our examples implements all three, so it is a functor, an applicative, and a monad.

What is the difference between the three? Functors: you apply a function to a wrapped value using `fmap` or `<\$>`.

Applicatives: you apply a wrapped function to a wrapped value using `apply` or `<*>`.

Monads: you apply a function that returns a wrapped value, to a wrapped value using `bind` or `>>=`.

So, dear friend (I think we are friends by this point), I think we both agree that monads are easy and a SMART IDEA(tm). Now that you've wet your whistle on this guide, why not pull a Mel Gibson and grab the whole bottle. Check out LYAH's section on Monads. There's a lot of things I've glossed over because Miran does a great job going in-depth with this stuff.

If you feel there is something I could change to improve this translation to ReasonML please let me know.

Thanks again to Aditya Bhargava for writing the original version of this post ❤️