# understanding higher-kinded types

*Kinds* are the types of types.

Okay, let me take a step back:

## proper types

*Values* have *types*.
Some typical types are `Integer`

, `Bool`

, and `String`

.
Values of type `Integer`

include `0`

, `1`

, `42`

.

Just as values can be grouped into types, types can be grouped into kinds.

`Bool`

and `Integer`

are among the simplest types.
We say that these types have kind `Type`

.^{1}
`Type`

is the kind containing all *populated* types —
those types which have values.

Types of kind `Type`

are the only types
which can be thought of as sets of values,
like `{true, false}`

or `{..., -2, -1, 0, 1, 2, ...}`

.
Populated types are also called *proper types*.
^{2}

## first-order types

There are also values of type `List(Integer)`

^{3}
such as `[1,2,3]`

and `[]`

,
so `List(Integer)`

has kind `Type`

.
But there are no values of type `List`

!

`List`

^{4} is not a proper type.
In a certain sense, it needs another (proper) type to complete it.
We can think of `List`

as a *function*
whose domain and codomain are both proper types.
`List`

has kind `Type -> Type`

.

Other examples of types with kind `Type -> Type`

are `Set`

and `Maybe`

^{5}.
Together, these are *first-order* types.

`Pair`

, `HashMap`

^{6}, and `Either`

^{7}
are not proper types either —
they are functions
whose domains are pairs of types
and whose codomains are types.
In other words, they have kind `(Type,Type) -> Type`

.
^{8}

There are no values of type `HashMap`

,
nor any of type `HashMap(String)`

,
but there are values of type `HashMap(String,Integer)`

such as `{ "hello": 2 , "hi": 3 }`

.
^{9}

Types which require any number
of other proper types as arguments
to form proper types themselves
are *first-order* types.
First-order types effectively allow us
to abstract over proper types.

## beyond first-order

But first-order types do not allow us to abstract
over other first-order types.
There is no such thing as a `List(List)`

.

Believe it or not, there are some (useful!) types
which require first-order types to complete them.
These types have kinds such as `(Type -> Type) -> Type`

and are called *higher-order* or *higher-kinded* types.

Examples include Foldable, Traversable, Functor, and Monad.

Kinds and first-order types can help us understand type-classes (or generics) as a logical extension of the type system.

Higher-kinded types take that a step further and include first-order types in our generics. They provide the means to abstract over types which themselves abstract over types.

*With thanks to hboo
for revising an earlier draft of this post.*

If you find it confusing that

`Type`

is the name of the kind containing all the types, remember that`Integer`

is the name of the type containing all the integers!↩︎There’s a minor technical difference between a proper type and a populated type. A populated type has kind

`Type`

and there is at least one value of that type. A proper type meets the first requirement, but not necessarily the second.`Void`

is a type with no values (an empty set), but it still has kind`Type`

. It is a proper type that is not populated.↩︎spelled with angle brackets in many languages, as in

`List<Integer>`

↩︎or

`Array`

↩︎or

`Optional`

↩︎or

`Dict`

↩︎or

`Result`

↩︎Also in this gang is

`->`

(function) — it has kind`(Type,Type) -> Type`

because it requires two types to complete it. In other words, there are no values of type`->`

, nor are there values of type`String ->`

or`-> Integer`

.There

*are*values of type`String -> Integer`

, such as the`length`

function where`length(s)`

is the number of characters in`s`

.Expressing functions using the

`->`

syntax can hide the fact that`Function`

is really just a type of kind`(Type,Type) -> Type`

like any other. You can think of`String -> Integer`

as`Function(String,Integer)`

if you prefer.↩︎In functional programming languages, these kinds are often curried. Instead of

`(Type,Type) -> Type`

,`HashMap`

might have kind`Type -> (Type -> Type)`

. A value would have type`HashMap(String)(Integer)`

.↩︎

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