Haskell Functors

Introduction to Functors

Explanation of Functor. A new Functor, f b, can be made from another Functor f a by transforming all of its value(s), whilst leaving the structure of f unchanged.

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

Its minimal complete definition is fmap.

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-- create a `f b` by calling function on each value in `f a`
fmap :: (a -> b) -> f a -> f b
-- infix form of `fmap`
(<$>) :: (a -> b) -> f a -> f b

-- create a `f a` by replacing all values in `f b` by `a`
(<$) :: a -> f b -> f a

-- almost the same, by reserving argument positions
($>) :: f a -> b -> f b

-- ignoring the values in `f a`.
void :: Functor f => f a -> f ()

Functor Laws

Types of Functor class must satisfy $2$ constraints:

1. Identity. When performing the mapping operation, if the values in the functor are mapped to themselves, the result will be an unmodified functor.

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fmap id = id

2. Composition.

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