How to interpret bind/>>= of the function instance?

Question

I\'m trying to improve my understanding of Applicatives and Monads by implementing their function instances in Javascript. My knowledge of Haskell

Answer 1

A few footnotes to Lee's answer:

However, the bind function looks pretty weird. Why are f and g nested the other way around?

Because bind is backwards. Compare (>>=) and its flipped version (=<<):

(>>=) :: Monad m => m a -> (a -> m b) -> m b
(=<<) :: Monad m => (a -> m b) -> m a -> m b

Or, in your specific example:

(>>=) :: (r -> a) -> (a -> (r -> b)) -> (r -> b)
(=<<) :: (a -> (r -> b)) -> (r -> a) -> (r -> b)

While in practice we tend to use (>>=) more often than (=<<) (because of how (>>=), syntactically speaking, lends itself well to the kind of pipeline monads are often used to build), from a theoretical point of view (=<<) is the most natural way of writing it. In particular, the parallels and differences with fmap/(<$>) and (<*>) are much more obvious:

(<$>) :: Functor f     =>   (a -> b) -> f a -> f b
(<*>) :: Applicative f => f (a -> b) -> f a -> f b
(=<<) :: Monad f       => (a -> f b) -> f a -> f b

When I apply apply/bind with an unary and a binary function, they yield the same result. This doesn't make much sense.

That is an accidental fact about the function instances. Let's put the specialised signatures side by side:

(<*>) :: (r -> (a -> b)) -> (r -> a) -> (r -> b)
(=<<) :: (a -> (r -> b)) -> (r -> a) -> (r -> b)

Monad goes beyond Applicative by providing the means to determine the next effect according to previous results (as opposed to "previous effect" -- Applicative can do that already). The effect, in this case, consists of a function that generates values given an argument of type r. Now, since functions with multiple arguments (i.e. functions that return functions) can be flipped, it happens that there is no significant difference between (r -> (a -> b)) and (a -> (r -> b)) (flip can trivially change one into the other), which makes the Monad instance for (->) r entirely equivalent to the Applicative one.