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  • Functors can only be defined over types that are [*] => *
  • Laws:
    • Identity law (identity function): map(fa, identity) = fa
    • Composition: map(map(fa,g), f) = map(fa, g andThen f) or andThen f) =
  • Functors compose very well
    • product of two functors is a functor
    • Sum of two functors is a functor
    • Nested functors is a functor
  • Functor preserves the structure (container type). For an example a list order can’t change.
  • Functor doesn’t need to be a data structure, it could be a function. Lots of data structures are functors but not all functors are data structures.
  • Functors doesn’t allow you to combine programs but only to change the return type.
trait functor[F[_]] {
  def map[A, B](fa: F[A])(f: A => B): F[B]

We can think of functors as a type of a programming language. For a functor F[_]

  • F: A sum type of different terms
  • F[A]: is a program in F. That is, its instructions in the F sum type. This program may halt of produce one or more A value(s).
  • Map: is the ability to change the output of a parser by replacing the value captured by the program a by f(a), where f: A => B


  • Option
    • None corresponds to a halt
    • Some Emits a value
  • List
    • Nil corresponds to a halt.
    • Cons(head, tail) recursive: Emits and then continue to the next element.


  • Nothing has a poly-kinded… that is it satisfies [*] => * and * and [*, *] => *, etc.


  1. Functional Scala by John A. De Goes (Toronto Edition)
  2. Scalaz Functor