Higher-Kinded Types (and Partial Application)

This page explains the “advanced” type shape behind Functor, Applicative, and Monad.

What is a “type constructor”?

Some types take type parameters:

• List a
• Option a
• Result a e

The bare names List and Option are type constructors (they still need an a).

In informal kind notation:

• List : * -> *
• Option : * -> *
• Result : * -> * -> *

Why Functor talks about f a

The class is defined as:

class Functor f
  map : (a -> b) -> f a -> f b

f here stands for a unary type constructor like List or Option.

Result is binary — so how can it be a Functor?

The prelude has an instance:

instance Functor (Result e)
  map = prim_map

Result e means: “fix the error type to e, leaving one type parameter for the Ok value”. Written fully (with both parameters), that’s Result a e.

So Result e behaves like a unary type constructor:

• Result a e is “a result with Ok type a and Err type e”
• map transforms the Ok value and leaves Err alone

Recognizing partial application in types

Whenever you see something like (Result e) or (Either e) in other languages, think:

“We pinned one type parameter to turn a multi-parameter type into a unary constructor.”

This idea shows up again for Applicative (Result e) and Monad (Result e).