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map, (but called fmap in Haskell 98) actually comes from the definition of a functor

That is, if f is a functor, and we are given a function of type (a -> b), and a container of type (f a), we can get a new container of type (f b). This is expressed in the type of fmap: fmap :: (Functor f) => (a -> b) -> f a -> f b If you will give me a blueberry for each apple I give you (a -> b), and I have a box of apples (f a), then I can get a box of blueberries (f b). Every monad is a functor.

The second method, return, is specific to monads. If m is a monad, then return takes an element of type a, and gives a container of type (m a) with that element in it. So, its type in Haskell is return :: (Monad m) => a -> m a If I have an apple (a) then I can put it in a box (m a).

takes a container of containers m (m a), and combines them into one m a in some sensible fashion. Its Haskell type is join :: (Monad m) => m (m a) -> m a

If I have a box of boxes of apples (m (m a)) then I can take the apples from each, and put them in a new box (m a).

bind or extend, which is commonly given the symbol (>>=)

The function that does this for any monad in Haskell is called liftM -- it can be written in terms of return and bind as follows:

liftM :: (Monad m) => (a -> b) -> m a -> m b liftM f xs = xs >>= (return . f) -- take a container full of a's, to each, apply f, -- put the resulting value of type b in a new container, -- and then join all the containers together.

Well, in Haskell, IO is a monad.

Lists are most likely the simplest, most illustrative example

The reason that bind is so important is that it serves to chain computations on monadic containers together.

You might notice a similarity here between bind and function application or composition, and this is no coincidence.

What bind does is to take a container of type (m a) and a function of type (a -> m b). It first maps the function over the container, (which would give an m (m b)) and then applies join to the result to get a container of type (m b). Its type and definition in Haskell is the

xs >>= f = join (fmap f xs)

bind (>>=)