Functor (functional programming) explained

In functional programming, a functor is a design pattern inspired by the definition from category theory that allows one to apply a function to values inside a generic type without changing the structure of the generic type. In Haskell this idea can be captured in a type class:

class Functor f where fmap :: (a -> b) -> f a -> f b

This declaration says that any type of Functor must support a method fmap, which maps a function over the element(s) of the Functor.

Functors in Haskell should also obey functor laws,^[1] which state that the mapping operation preserves the identity function and composition of functions:

fmap id = idfmap (g . h) = (fmap g) . (fmap h)

(where . stands for function composition).

In Scala a trait can be used:

trait Functor[F[_]]

Functors form a base for more complex abstractions like Applicative Functor, Monad, and Comonad, all of which build atop a canonical functor structure. Functors are useful in modeling functional effects by values of parameterized data types. Modifiable computations are modeled by allowing a pure function to be applied to values of the "inner" type, thus creating the new overall value which represents the modified computation (which might yet to be run).

Examples

In Haskell, lists are a simple example of a functor. We may implement as

fmap f [] = []fmap f (x:xs) = (f x) : fmap f xs

A binary tree may similarly be described as a functor:

data Tree a = Leaf | Node a (Tree a) (Tree a)instance Functor Tree where fmap f Leaf = Leaf fmap f (Node x l r) = Node (f x) (fmap f l) (fmap f r)

If we have a binary tree and a function, the function will apply to every element of . For example, if is, adding 1 to each element of can be expressed as .^[2]

See also

• Functor in category theory
• Applicative functor, a special type of functor

External links

• Section about Functor in Haskell Typeclassopedia
• Chapter 11 Functors, Applicative Functors and Monoids in Learn You a Haskell for Great Good!
• Documentation for Functor in Cats library
• Section about Functor in lemastero/scala_typeclassopedia

Notes and References

1. Web site: Yorgey . Brent . Functor > Laws . HaskellWiki . 17 June 2023.
2. Web site: Functors . Functional Pearls . University of Maryland . 12 December 2022.