Monad
can pass Just [1,2]
, which is a different type from what the original length
function takes, to >>= return . length
.
Just [1,2] >>= return . length
Can I say that Monad
makes it possible to see Maybe [a]
as isomorphic with [a]
on length using (>>=, return)
? (Of course they are not really isomorphic.)
Can I choose the term "isomorphic" this situation?
What your example ultimately illustrates is that Maybe
is a functor: if you have some f :: a -> b
, you can use fmap
to turn it into fmap f :: Maybe a -> Maybe b
in a way that preserves identities and composition. Monads are functors, with \f m -> m >>= return . f
being the same as fmap f m
. In your case, we have the length
function being transformed by the Maybe
functor.
can I choose term "isomorphic" this situation?
Not really. fmap
for Maybe
is not an isomorphism. An isomorphism requires there being a two-sided inverse that undoes it, which in this case would be something like:
unFmapMaybe :: (Maybe a -> Maybe b) -> (a -> b)
-- For it to be a two-sided inverse to `fmap`, we should have:
unFmapMaybe . fmap = id
fmap . unFmapMaybe = id
However, there are no (Maybe a -> Maybe b) -> (a -> b)
functions, as there is no way to obtain a b
result if the input Maybe a -> Maybe b
function gives out a Nothing
. While there are specific functors whose fmap
is an isomorphism (Identity
is one example), that is not the case in general.