[Functional Programming] Working with two functors（Applicative Functors）-- Part2 --liftAN

Let's examine a pointfree way to write these applicative calls. Since we know map is equal to of/ap, we can write generic functions that will ap as many times as we specify:

const liftA2 = curry((g, f1, f2) => f1.map(g).ap(f2));

const liftA3 = curry((g, f1, f2, f3) => f1.map(g).ap(f2).ap(f3));

// liftA4, etc

Let's see the previous examples written this way:

const profile = name => email => `${name}__${email}`;
const safeProfile = liftA2(profile);
const res1 = safeProfile(prop('name', user), prop('email', user)); // John Doe__blurp_blurp

liftA2(add, Maybe.of(2), Maybe.of(3));
// Maybe(5)

liftA2(renderPage, Http.get('/destinations'), Http.get('/events'));
// Task('<div>some page with dest and events</div>')

liftA3(signIn, getVal('#email'), getVal('#password'), IO.of(false));
// IO({ id: 3, email: 'gg@allin.com' })

liftAN: Lift a curry function into a Functor context, which will be define later;  

liftA2(add, Maybe.of(2), Maybe.of(3)); Maybe will be the Functor context for 'add' function which has been lifted

Laws:

Identity

// identity
A.of(id).ap(v) === v;

For example:

const v = Identity.of('Pillow Pets');
Identity.of(id).ap(v) === v;

Homomorphism

// homomorphism
A.of(f).ap(A.of(x)) === A.of(f(x));

A homomorphism is just a structure preserving map. In fact, a functor is just a homomorphism between categories as it preserves the original category's structure under the mapping.

A quick example:

Either.of(toUpperCase).ap(Either.of('oreos')) === Either.of(toUpperCase('oreos'));

Interchange

The interchange law states that it doesn't matter if we choose to lift our function into the left or right side of ap.

// interchange
v.ap(A.of(x)) === A.of(f => f(x)).ap(v);

Here is an example:

const v = Task.of(reverse);
const x = 'Sparklehorse';

v.ap(Task.of(x)) === Task.of(f => f(x)).ap(v);

Composition

// composition
A.of(compose).ap(u).ap(v).ap(w) === u.ap(v.ap(w));

const u = IO.of(toUpperCase);
const v = IO.of(concat('& beyond'));
const w = IO.of('blood bath ');

IO.of(compose).ap(u).ap(v).ap(w) === u.ap(v.ap(w));

Examples:

const safeAdd = curry((a, b) => Maybe.of(add).ap(a).ap(b));
const safeAdd = liftA2(add);

const localStorage = {  
  player1: { id:1, name: 'Albert' },  
  player2: { id:2, name: 'Theresa' },  
};  

// getFromCache :: String -> IO User  
const getFromCache = x => new IO(() => localStorage[x]);  

// game :: User -> User -> String  
const game = curry((p1, p2) => `${p1.name} vs ${p2.name}`);
// startGame :: IO String
const startGame = liftA2(game, getFromCache('player1'), getFromCache('player2'));