Monads

Data structures with map and flatMap seem to be quite common. In fact there’s a name that describes this class of a data structures together with some algebraic laws that they should have. They are called monads.

A monad M is a parametric type M[T] with two operations, flatMap and unit, that have to satisfy some laws.

trait M[T] {
    def flatMap[U](f: T => M[U]): M[U]
}

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

In the literature, flatMap is more commonly called bind.

Examples:

• List is a monad with unit(x) = List(x)
• Set is monad with unit(x) = Set(x)
• Option is a monad with unit(x) = Some(x)
• Generator is a monad with unit(x) = single(x)

flatMap is an operation on each of these types, whereas unit in Scala is different for each monad.

Maps and Monads

map can be defined for every monad as a combination of flatMap and unit:

m map f == m flatMap (x => unit(f(x)))
// OR
m map f == m flatMap (f andThen unit)

Monad Laws

To qualify as a monad, a type has to satisfy three laws:

1. Associativity:

m flatMap f flatMap g == m flatMap (x => f(x) flatMap g)

2. Left unit:

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

3. Right unit:

m flatMap unit == m

Checking Monad Laws