#fmap

I keep confusing myself with these, and references to List and Maybe as Functors is also confusing so I want to clear this up for myself (and provide a reference for myself in the future in case I forget).

Its not untrue that they’re Functors but they are also Not Functors sort of. They’re more like domains that can be lifted to by Functors if that makes more sense? But since there isn’t a name for these Functors, we call it a List Functor or a Maybe Functor I suppose.

So the Functor Law (from the Haskell Wiki - a Quite Reliable Source) says that Functors must follow two Laws. Identity preservations (fairly trivial), and composition which is a little more complicated. I think Id is fairly trivial so I’m going to ignore Id.

First things first, basics. fmap is defined as fmap :: (a -> b) -> f a -> f b. The f is referring to the (a -> b)

So when we talk about List or Maybe being functors what we mean is that we can take some Function f defined as f :: a -> b and then call fmap f (Maybe a) and we will get Maybe b out of it, which means that we can do a -> b without taking it out of the Maybe “wrapper”. In a purely category theory sense, a functor is the transformation the normal types a and b to the wrapped types Maybe a and Maybe b and since its a functor the morphism (function) is also preserved into the Maybe domain, it turns into a Maybe a -> Maybe b.

Now, lets talk about composition (Id is simple, Id should be the same in all domains, if its a -> a then it certainly should be Maybe a -> Maybe b). Composition says composition of morphisms is preserved so

fmap (f . g)  ==  fmap f . fmap g

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