Functors Compose, Monads Do Not
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Functor Composition
Let’s start with a refresher of map
:
map :: forall a b. (a -> b) -> f a -> f b
In other words, map takes a function a -> b
and gives us a function f a -> f b
. For that reason, we can take any two nested functors (e.g. Array
and Maybe
) and run a function on the nested values by putting two map
s together:
v :: Array (Maybe Int)
v = [Just 1, Nothing, Just 3]f1 :: Int -> String
f1 = showmain :: Effect Unit
main = do
logShow $ map (map f1) v-- [(Just "1"),Nothing,(Just "3")]
Monad Composition
This time we want to take a look at bind
:
bind :: forall a b. m a -> (a -> m b) -> m b
If we tried to compose the same way we did with functors, we would notice the code does not compile:
v :: Array (Maybe Int)
v = [Just 1, Nothing, Just 3]f2 :: Int -> Array (Maybe String)
f2 i = [Just $ show i]main :: Effect Unit
main = do
logShow $ bind v (\x -> bind x f2) -- DOES NOT COMPILE!!
The problem here is in the nested bind
:
bind v (\x -> bind x f2)
^ Maybe Int
^ Int -> Array (Maybe String)
In fact, Maybe Int -> (Int -> Array (Maybe String)) -> ??
is not what bind
expects: the first argument seems to indicate that m
is Maybe
but the second seems to indicate that m
is Array
. This does not compile since the monad m
is supposed to be the same.
To make the program compile we have to make use of a function (i.e. maybe
) specific to the monad we are dealing with (i.e. Maybe
):
main :: Effect Unit
main = do
-- logShow $ bind v (\x -> bind x f2) -- DOES NOT COMPILE!!
logShow $ bind v (maybe (pure Nothing) f2)
Or we could use the MaybeT
monad transformer:
v2 :: MaybeT Array Int
v2 = MaybeT [Just 1, Nothing, Just 3]f3 :: Int -> MaybeT Array String
f3 i = MaybeT [Just $ show i]main :: Effect Unit
main = do
--logShow $ bind v (\x -> bind x f2)
logShow $ bind v (maybe [Nothing] f2)
logShow $ runMaybeT $ bind v2 f3
Outro
I’ve blatantly copied the content of this blog post out of a talk by Tony Morris. So be sure to check the original stuff out!
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