Functor module:Data package:base

A type f is a Functor if it provides a function fmap which, given any types a and b, lets you apply any function of type (a -> b) to turn an f a into an f b, preserving the structure of f.

Examples

>>> fmap show (Just 1)  --  (a   -> b)      -> f a       -> f b
Just "1"                --  (Int -> String) -> Maybe Int -> Maybe String

>>> fmap show Nothing   --  (a   -> b)      -> f a       -> f b
Nothing                 --  (Int -> String) -> Maybe Int -> Maybe String

>>> fmap show [1,2,3]   --  (a   -> b)      -> f a       -> f b
["1","2","3"]           --  (Int -> String) -> [Int]     -> [String]

>>> fmap show []        --  (a   -> b)      -> f a       -> f b
[]                      --  (Int -> String) -> [Int]     -> [String]

The fmap function is also available as the infix operator <$>:

>>> fmap show (Just 1) --  (Int -> String) -> Maybe Int -> Maybe String
Just "1"

>>> show <$> (Just 1)  --  (Int -> String) -> Maybe Int -> Maybe String
Just "1"

A type f is a Functor if it provides a function fmap which, given any types a and b lets you apply any function from (a -> b) to turn an f a into an f b, preserving the structure of f. Furthermore f needs to adhere to the following:

• Identity fmap id == id
• Composition fmap (f . g) == fmap f . fmap g

Note, that the second law follows from the free theorem of the type fmap and the first law, so you need only check that the former condition holds. See https://www.schoolofhaskell.com/user/edwardk/snippets/fmap or https://github.com/quchen/articles/blob/master/second_functor_law.md for an explanation.