A cheat-sheet for Haskell & PureScript typeclasses

One-page overview of a bunch of different Haskell & PureScript typeclasses. Starting with the PureScript Class Hierarchy diagram, and similar on the Haskell Typeclassopedia. Very much a work in progress: info and terminology may be completely wrong!


Functor

Map a value inside a context to a new value:

class Functor f where
  fmap :: (a -> b) -> f a -> f b

Applicative

With a function (or, more intuitively, a partial computation) inside a context, map a value also inside the same context:

class Functor f => Applicative f where
  (<*>) :: f (a -> b) -> f a -> f b

Also provide a way to lift a value into a context:

  pure  :: a -> f a

Usage

For instance, how could we sum Just 2 and Just 3? 1

> :t fmap (+)
fmap (+) :: (Functor f, Num a) => f a -> f (a -> a)
> :t fmap (+) (Just 2)
fmap (+) (Just 2) :: Num a => Maybe (a -> a)

Ah, now we can use <*>:

fmap (+) (Just 2) <*> (Just 3)

So when you have a function a -> b -> c, partially apply it, and lift into the functor, you end up with f (b -> c), then to apply it to a second parameter of type b, you use <*>, and end up with f c. Thus:

<*> is just function application within a computational context.2

Monad

Use the result of a previous computation as input into another computation, i.e. sequence computations.

One consequence is that a sequence may not preserve the structure of an input. An example: when under Maybe, functors and applicatives can’t change the result from Just a to Nothing, since none of the parameters return f b, they only return b. You need to “upgrade” to a monad to be able to use a function like a -> m b.


Semigroup

Set S with a binary operation <> that combines two elements of S into another element of S:

class Semigroup a where
  (<>) :: a -> a -> a

<> must be associative.

Monoid

Extension of a Semigroup with an identity element i, such that s <> i == i <> s == s.

Semigroupoid

Set of objects with composable morphisms. E.g. like a category, but without requiring an identity element.

class Semigroupoid a where
  compose :: forall b c d. a c d -> a b c -> a b d

(->) is a semigroupoid, with (.) as compose: (.) :: (b -> c) -> (a -> b) -> (a -> c).

Category

Semigroupoid with an identity element.

To be less facetious, a category is a set of objects, composable morphisms, and an identity element, such that id <<< p = p <<< id = p

Group

A Group is a Monoid with inverses. Numbers form a group with + as the inverse of a value a is -a:

a + -a = zero

Semiring

types that support addition and multiplication.

Ring

Semiring that also supports subtraction.

Arrow

a value of type b `arr` c can be thought of as a computation which takes values of type b as input, and produces values of type c as output.

  1. Haskell/Applicative functors on WikiBooks

  2. Typeclassopedia: §4.1 [Applicative] Definition