Haskell Language Category Theory Haskell Applicative in terms of Category Theory
Example
A Haskell's Functor allows one to map any type a (an object of Hask) to a type F a and also map a function a -> b (a morphism of Hask) to a function with type F a -> F b. This corresponds to a Category Theory definition in a sense that functor preserves basic category structure.
A monoidal category is a category that has some additional structure:
- A tensor product (see Product of types in Hask)
- A tensor unit (unit object)
Taking a pair as our product, this definition can be translated to Haskell in the following way:
class Functor f => Monoidal f where
mcat :: f a -> f b -> f (a,b)
munit :: f ()
The Applicative class is equivalent to this Monoidal one and thus can be implemented in terms of it:
instance Monoidal f => Applicative f where
pure x = fmap (const x) munit
f <*> fa = (\(f, a) -> f a) <$> (mcat f fa)
