G'day all.
Quoting David Menendez
Until recently, I've had difficulty relating the concept of a "functor" from Category Theory to the class Functor in Haskell.
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We can interpret Haskell as a category with types as objects and functions as morphisms. For example, isDigit :: Char -> Bool.
So, for some Functor f, the type constructor f corresponds to F[ob], and fmap to F[mor]. For any function g :: a -> b, fmap g :: f a -> f b, and fmap (g . h) = fmap g . fmap h.
Correct. Technically, Functor should probably be called Endofunctor, because it's actually a functor from the category Haskell to the category Haskell. That would probably just confuse things more, though. :-)
For two functors F, G: C -> D, there is a natural transformation tau from F to G, if for every object a in C, there is morphism tau[a]: F(a) -> G(a) in D.
In Haskell, natural transformations are polymorphic functions, tau :: f a -> g a. For example, maybeToList :: Maybe a -> [a].
Not quite. A natural transformation is a mapping between _functors_. In this case, maybeToList is a natural transformation between Maybe and []. Expressing this as a typeclass is left as an exercise, but here's a start: class (Functor f, Functor g) => NaturalTransformation ... where ...
A monad is a functor, F: C -> C, plus two natural transformations mu: F^2 -> F and eta: I -> F (I being the identity functor for C).
In Haskell, F[ob] becomes the type constructor, F[mor] becomes fmap (or liftM), mu becomes join :: f (f a) -> f a, and eta becomes return :: a -> f a.
Yup! Handily, all Haskell Functors are endofunctors, so Haskell monads don't need the extra constraint that F must be an endofunctor. Cheers, Andrew Bromage