[Functional Programming] Monad

Before we introduce what is Monad, first let's recap what is a pointed functor:

A pointed functor is a Functor with .of() method

Why pointed Functor is imporant? here

OK, now, let's continue to see some code:

const mmo = Maybe.of(Maybe.of('nunchucks'));
// Maybe(Maybe('nunchucks'))

We don't really want nested Functor, it is hard for us to work with, we need to remember how deep is the nested Functor.

To solve the problem we can have a new method, call '.join()'.

mmo.join();
// Maybe('nunchucks')

What '.join()' does is just simply reduce one level Functor.

So how does implememation of 'join()' looks like?

Maybe.prototype.join = function join() {
  return this.isNothing() ? Maybe.of(null) : this.$value;
};

As you can see, we just return 'this.$value', instead of put the value into Maybe again.

With those in mind, let's define what is Monad!

Monads are pointed functors that can flatten

Let's see a example, how to use join:

// join :: Monad m => m (m a) -> m a
const join = mma => mma.join();

// firstAddressStreet :: User -> Maybe Street
const firstAddressStreet = compose(
  join,
  map(safeProp('street')),
  join,
  map(safeHead), safeProp('addresses'),
);

firstAddressStreet({
  addresses: [{ street: { name: 'Mulburry', number: 8402 }, postcode: 'WC2N' }],
});
// Maybe({name: 'Mulburry', number: 8402})

For now, each map opreation which return a nested map, return call 'join' after.

Let's abstract this into a function called chain.

// chain :: Monad m => (a -> m b) -> m a -> m b
const chain = curry((f, m) => m.map(f).join());

// or

// chain :: Monad m => (a -> m b) -> m a -> m b
const chain = f => compose(join, map(f));

Now we can rewrite the previous example which .chain():

// map/join
const firstAddressStreet = compose(
  join,
  map(safeProp('street')),
  join,
  map(safeHead),
  safeProp('addresses'),
);

// chain
const firstAddressStreet = compose(
  chain(safeProp('street')),
  chain(safeHead),
  safeProp('addresses'),
);

To get a feelings about chain, we give few more examples:

// getJSON :: Url -> Params -> Task JSON
getJSON('/authenticate', { username: 'stale', password: 'crackers' })
  .chain(user => getJSON('/friends', { user_id: user.id }));
// Task([{name: 'Seimith', id: 14}, {name: 'Ric', id: 39}]);

// querySelector :: Selector -> IO DOM
querySelector('input.username')
  .chain(({ value: uname }) => querySelector('input.email')
  .chain(({ value: email }) => IO.of(`Welcome ${uname} prepare for spam at ${email}`)));
// IO('Welcome Olivia prepare for spam at olivia@tremorcontrol.net');

Maybe.of(3)
  .chain(three => Maybe.of(2).map(add(three)));
// Maybe(5);

Maybe.of(null)
  .chain(safeProp('address'))
  .chain(safeProp('street'));
// Maybe(null);

Theory

The first law we'll look at is associativity, but perhaps not in the way you're used to it.

// associativity
compose(join, map(join)) === compose(join, join);

These laws get at the nested nature of monads so associativity focuses on joining the inner or outer types first to achieve the same result. A picture might be more instructive:

The second law is similar:

// identity for all (M a)
compose(join, of) === compose(join, map(of)) === id;

It states that, for any monad M, of and join amounts to id. We can also map(of) and attack it from the inside out. We call this "triangle identity" because it makes such a shape when visualized:

Now, I've seen these laws, identity and associativity, somewhere before... Hold on, I'm thinking...Yes of course! They are the laws for a category. But that would mean we need a composition function to complete the definition. Behold:

const mcompose = (f, g) => compose(chain(f), g);

// left identity
mcompose(M, f) === f;

// right identity
mcompose(f, M) === f;

// associativity
mcompose(mcompose(f, g), h) === mcompose(f, mcompose(g, h));

They are the category laws after all. Monads form a category called the "Kleisli category" where all objects are monads and morphisms are chained functions. I don't mean to taunt you with bits and bobs of category theory without much explanation of how the jigsaw fits together. The intention is to scratch the surface enough to show the relevance and spark some interest while focusing on the practical properties we can use each day.

More detail