Functors in Programming

Functor maps

• every object in one category C, for example, to one selected object in another category D,
• it also maps morphism in source category to the target category.

It's easy to understand how morphism is mapped, for example, f: a -> b mapped to Ff : F a -> F b So how Functor maps object a to F a ? This is implemented through Type constructor, i.e:

data Maybe a = Nothing | Just a

Here Maybe without type parameters is not a type, it's a type constructor, and is** type-level function** when applies to another type, it creates a new type, Maybe type constructor(type level function) maps a to Maybe a Haskell uses a typeclass to express functoriality:

//we say fmap lifts a function
class Functor f where
    fmap :: (a->b) -> (f a -> f b)

Compiler knows that f is a type constructor because it is applied to types as in f a, f b