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A functional programming idiom for mapping over containers

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The term "functor" originates from category theory, but this will be of no further concern in this description. In functional programming, "functor" refers to a programming idiom for mapping over contains or compound data. Functors have been popularized by Language:Haskell.

In Haskell, functors are programmed and used with the help of the type class Functor which is parametrized by a type constructor for the actual container type:

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

The type constructor parameter f is the placeholder for the actual container type. The fmap function (for "functorial map") is the principle operation of a functor: parametrized by a function for mapping container elements of type a to elements of type b, it provides a mapping at the level of the container types, from f a to f b. Algebraically, the following properties are required for any functor (given in Haskell notation):

fmap id = id
fmap f . fmap g = fmap (f . g)

The following Functor instance turns lists into a functor:

instance Functor []
  where
    fmap = map

Thus, the folklore map function for list processing is a particular example of the notion of functorial map.

Here is another Functor instance turning the Maybe type constructor into a functor.

instance Functor Maybe
  where
    fmap _ Nothing = Nothing
    fmap f (Just x) = Just (f x)

See also the concept of rose trees for more complicated examples of functors.

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