A simple encoding of higher-kinded types in TypeScript

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TypeScript doesn't really support higher-kinded types yet, but various attempts have been made to simulate them (see related work at the bottom). This project is one such idea, which attempts to solve the problem via conditional types.
The idea is that a type which logically depends on a type constructor (rather than a simple type) just takes a regular type variable, and then uses the $ operator to "apply" that variable to other types. For example, here's how we would write the Functor type class as defined by static-land:
interface Functor<T> {
  map: <A, B>(f: (x: A) => B, t: $<T, [A]>) => $<T, [B]>;

Then, supposing we have a Maybe type
type Maybe<A> = { tag: 'none' } | { tag: 'some'; value: A };
const none: Maybe<never> = { tag: 'none' };
const some = <A>(value: A): Maybe<A> => ({ tag: 'some', value });

We can define a Functor instance for it like so:
const MaybeFunctor: Functor<Maybe<_>> = {
  map: (f, t) => t.tag === 'none' ? none : some(f(t.value)),

Notice that we are supplying the Maybe type constructor with the placeholder type _; this causes it to be come a fully saturated type so that we can pass it to Functor, but with all former occurrences of the type parameter clearly marked, so that they can be re-substituted using the $ operator. A type application $<T, S> then recursively walks the tree of type T, substituting any placeholders _<N> it finds with the corresponding argument type S[N]. _ is shorthand for _<0>, and there are also placeholder aliases _0 = _<0>, _1 = _<1>, etc.
Take a look at the tests for more examples.

Type classes and instance factories

This package defines a set of interfaces corresponding to the the type classes of the static-land spec, as well as factory functions for producing instances thereof. For example, there is a Monad interface as well as a Monad function. The function takes as arguments only the minimum data needed (of and chain) to produce an implementation of the full Monad interface (which includes other derived methods like map, ap and join). So again using the Maybe type above, we can construct a Monad instance like so:
const MaybeMonad = Monad<Maybe<_>>({
  of: some,
  chain: (f, t) => t.tag === 'none' ? none : f(t.value),

// Use the `map` method, which we didn't have to define:
expect( => n + 1, some(42))).toBe(some(43));

Abstracting over kinds of higher arity

As alluded to above, the type S of $<T, S> is a tuple of types [S0, S1, ..., SN] to be substituted for the corresponding placeholders _0, _1, ..., _<N>. This allows us to define for example Bifunctor in a straightforward way:
interface Bifunctor<T> {
  bimap: <A, B, C, D>(f: (x: A) => B, g: (x: C) => D, t: $<T, [A, C]>) => $<T, [B, D]>;
  // ...

and a Bifunctor instance for Either using placeholders _0 and _1:
type EitherBifunctor: Bifunctor<Either<_0, _1>> = {
  bimap: (f, g, t) => (t.tag === 'left' ? left(f(t.left)) : right(g(t.right))),

Fixing type parameters

Some trickiness arises when we want to ignore one or more of the parameters of a type constructor for the purpose of making it an instance of a type class. For example we can make a Monad out of Either by ignoring its first parameter and using the second parameter as the "hole" of the monad. The way to do this is to make a polymorphic instance creation function which can produce a Monad instance for any given left type L:
const RightMonad = <L>() => Monad<Either<L, _>>({
  pure: right,
  bind: (ma, f) => (ma.tag === 'left' ? ma : f(ma.right)),

Known limitations

The type application operator $ is able to transform most types correctly, including functions, however there are a few edge cases:
  • Polymorphic functions will get nerfed. For example:
```ts type Id = (x: A) => A type NerfedId = $>; // type NerfedId = (x0: {}) => {} `` Not what you wanted! Unfortunately TypeScript's conditional types don't currently allow analyzing and reconstructing the type parameters of a function ( might solve that). You can protect polymorphic types with Fixed` as well, although then they can't contain placeholders, so that's probably not of much use to you.
  • Tuples [A, B, ...] are transformed correctly up to size 10 (though we can add arbitrarily many more as needed), after which they will be transformed into an array (A | B | ...)[]. Correspondingly, functions of arity <= 10 are supported.
  • The join method of Monad must be supplied a type parameter for some reason... I've filed for this. The good news is it's still safe; you can't provide a wrong type argument without getting an error.

Related work

Other notable attempts to solve this problem: