Monoid, Functor, Applicative and Monad

Monoid, Functor, Applicative and Monad

Monoid

Given a Monoid trait Semigroup

trait Semigroup[M] {
  def append(a: M, b: M): M
  val zero: M
}

the following should hold:

append(a, append(b, c)) === append(append(a, b), c)
append(a, zero) = a
append(zero, a) = a

Monoid examples:

• Int with + and 0
• Int with * and 1
• Boolean with || and false
• A => A with compose and identity
• List[A] with ++ and Nil
• String with + and ""

Functor

Concept

Functor is a type class that defines how to apply a function to a value wrapped in a context(T).List, Option, Ethier, Try both are functor.

trait Functor[T[_], A] {
  def fmap[B](f: A => B): Functor[T, B]

  def id: T[A]
}

the Functor takes two type parameters, T[_] which is a generic type, and a type A one concrete example is:

//List as T, A as A
case class ListFunctor[A](val id: A, xs: List[A]) extends Functor[List, A] {
  def fmap[B](f: A => B): List[B] = ListFunctor(xs.map(f))
}

Functor laws:

• fmap id = id

  if we map the id function over a functor, the functor that we get back should be the same as the original functor

• for any functor F, the following should hold: fmap (f . g) F = fmap f (fmap g F)

  composing two functions and then mapping the resulting function over a functor should be the same as first mapping one function over the functor and then mapping the other one

Function is Functor:

Function composition:

Mapping a Function over a Function will produce a new Function(function composition), just like mapping a function over a List will produce a List, mapping a function over a Option will produce a Option.

Lifting:

Given a map function with type (A => B) => F[A] => F[B](F is a functor, it could be List, Option, or Ethier), we can think the map as a function which take a function (with type A => B) as parameter and return a new function just like the old one(with type F[A] => F[B]).

Applicative

Concept

Applicative is a type class that defines how to apply a function tf wrapped in a context T to a value wrapped in a context T.

trait Applicative[T[_], A] extends Functor[T, A] {
  def apply[B](f: T[A => B]): Applicative[T, B]
}

Monad

Concept

Monad is a type class Monad[T[_], A] that defines how to apply a function that returns a wrapped value A => T[B] to a wrapped value T[A].

trait Monad[T[_], A] extends Monoid[T, A] with Applicative[T, A] {
  def flatMap[B](f: A => T[B]): Monad[T, B]
}

Monad law:

• Left identity

  Given a value x and a function f, the following should hold:

unit(x) flatMap f = f(x)

• Right identity

Given a monad m, the following should hold:

m flatMap unit = m

• Composition

Given a monad m and two functions f and g, the following should hold:

m flatMap f flatMap g == m flatMap g flatMap f

A concrete Monad example

case class ListMonad[A](val list: List[A])  extends Monad[List, A] {
  //defined in Monoid
  override def append(values: List[A]): ListMonad[A] = ListMonad(list ++ values)

  //defined in Monoid
  override def id: List[A] = Nil

  //defined in Functor
  override def fmap[B](f: (A) => B): ListMonad[B] = ListMonad(list.map(f))

  //defined in Applicative
  override def apply[B](tf: List[(A) => B]): ListMonad[B] = ListMonad(list.map(tf.head))

  //defined in Monad
  override def flatMap[B](f: (A) => List[B]): ListMonad[B] = ListMonad(list.flatMap(f))
}