function-as-functor-applicative-monad.md

In Haskell, functions are instances of Functor, Applicative and Monad.

Very good article with a lot of details: https://eli.thegreenplace.net/2018/haskell-functions-as-functors-applicatives-and-monads/ .

TLDR;

Function as a Functor

g <$> f == \x -> g (f x) == g . f

What really happens is just composition. Basically you apply g on the return value of the function f.

Visually, we go from

      f
x --> _

to

      f     g
x --> _ --> _

Function as an Applicative

g <*> f == \x -> g x (f x)

You apply g onto two arguments, first x and second f x. I think of it as piping x first into f, and then piping x and f x into g.

Visually, we go from

      f
x --> _

to

            g
x --------> _
│     f
└---> _ --> _

Function as a Monad

f >>= g = \x -> g (f x) x

Same as with applicative, but flipped order of arguments that get passed to g.

Visually, we go from

      f
x --> _

to

      f     g
x --> _ --> _
└---------> _

Note that for Applicative and Monad g must be a function that takes at least two arguments, while for Functor it must take at least one argument.