I know that Functor typeclass is defined like so:
class Functor f where
fmap :: (a -> b) -> f a -> f b
In my course it says that this is how a list is made an instance of functor:
instance Functor [] where
fmap = map
Is [] a some kind of synonym for [a]?
Is
[] asome kind of synonym for[a]
Yes, precisely. There's no difference between the two types: the latter is only a pretty notation for the former.
In Haskell, all types follow the syntax T arg1 arg2 ... where T is some type constructor, but some of them also have pretty notations hard-coded in the language for human convenience. Here's a few:
a -> b means (->) a b
[a] means [] a
(a,b) means (,) a b
(a,b,c) means (,,) a b c
... ditto for other tuple types
Because of this, one can find a functor instance such as instance Functor ((->) a) whose fmap has type
(x -> y) -> ((->) a x) -> ((->) a y)
which can also be written
(x -> y) -> (a -> x) -> (a -> y)
Such fmap is just function composition, i.e., fmap = (.).