Monads

A monad M is a parametric type M[T] with 2 operations:

• flatMap
• unit

that have to satisfy some laws.

extension [T, U](m: M[T])
  def flatMap(f: T => M[U]): M[U]

def unit[T](x: T): M[U]

Examples of Monads

• List: unit(x) = List(x)
• Set: unit(x) = Set(x)
• Option: unit(x) = Some(x)

Monad Laws

• Associativity

  m.flatMap(f).flatMap(g) == m.flatMap(f(_).flatMap(g))
  

• Left unit

  unit(x).flatMap(f) == f(x)
  

• Right unit

  m.flatMap(unit) == m
  

Significance of the laws for For-comprehension

1. Associativity says that one can "inline" nested.

   for
     y <- for x <- m; y <- f(x) yield y
     z <- g(y)
   yield z
   
   // equivalent to
   
   for x <- m; y <- f(x); z <- g(y)
   yield z
   

2. Right unit says

   for x <- m yield x
   
   // equivalent to
   
   m