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).
Examples In Haskell, lists are a simple example of a functor. We may implement fmap as
A binary tree may similarly be described as a functor:
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.
See also Functor in category theory Applicative functor, a special type of functor
References
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
