What is Applicative Functor definition from the category theory POV?

Question

I was able to map Functor\'s definition from category theory to Haskell\'s definition in the following way: since objects of Hask are types, the functor F

Answer 1

You're right, Applicative translates less straightforwardly than Functor or Monad. But in essence, it is the class of monoidal functors:

class Functor f => Monoidal f where
  pureUnit :: f ()
  fzip :: f a -> f b -> f (a,b)

From that you can define – within Hask –

pure x = fmap (const x) pureUnit

and

fs <*> xs = fmap (uncurry ($)) $ fzip fs xs

See this answer for a full proof that Applicative and Monoidal are really equivalent.