Functor (functional programming)
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: This declaration says that any instance of Functor must support a method fmap, which maps a function over the elements of the instance. Functors in Haskell should also obey the so-called functor laws, which state that the mapping operation preserves the identity function and composition of functions: where . stands for function composition. In Scala a trait can instead be used: Functors form a base for more complex abstractions like applicative functors, monads, and comonads, 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 may have yet to run).
fmap (+1) to a binary tree of integers increments each integer in the tree by one.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 instance of Functor must support a method fmap, which maps a function over the elements of the instance.
Functors in Haskell should also obey the so-called functor laws,[1] which state that the mapping operation preserves the identity function and composition of functions:
fmap id = id
fmap (g . h) = (fmap g) . (fmap h)
where . stands for function composition.
In Scala a trait can instead be used:
trait Functor[F[_]] {
def map[A,B](a: F[A])(f: A => B): F[B]
}
Functors form a base for more complex abstractions like applicative functors, monads, and comonads, 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 may have yet to run).
Examples
[edit]In Haskell, lists are a simple example of a functor. We may implement fmap 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 tr :: Tree a and a function f :: a -> b, the function fmap f tr will apply f to every element of tr. For example, if a is Int, adding 1 to each element of tr can be expressed as fmap (+ 1) tr.[2]
See also
[edit]- Functor in category theory
- Applicative functor, a special type of functor
References
[edit]- ^ Yorgey, Brent. "Functor > Laws". HaskellWiki. Retrieved 17 June 2023.
- ^ "Functors". Functional Pearls. University of Maryland. Retrieved 12 December 2022.