Im a Haskell beginner and I'm still learning about Category Theory and its practical use in computer science.
I've spent last day watching couple lectures from Berkley's university about category theory, most of its content was showing a mathematical view of Rings, Semigroups, Groups, Magmas, Monoids, etc.
Hence, questions raised in my mind about monadic composition and kleisli category. Therefore, I would like to questions Haskell/Category Theory experts.
Is do notation a sort of monad composition?
Regards,
Pablo Parada
Do notation is just syntactic sugar for >>=. Code such as
do x <- a
b -- b is an expression possibly involving x
is desugared to
a >>= \x -> b
If you are studying monads in CT, you will probably find that they are being defined as functors with two natural transformations
unit :: a -> m a -- also known as η
join :: m (m a) -> m a -- also known as μ
while Haskell defines
return :: a -> m a
(>>=) :: m a -> (a -> m b) -> m b
Both presentations are equivalent. Indeed, unit and return are exactly the same thing. Instead, join can be expressed in terms of (>>=) as follows
join x = x >>= id
and, vice versa, (>>=) can be expressed in terms of join.
x >>= f = join (fmap f x)
above note that fmap takes a -> m b and m a to return m (m b), which is then flattened to m b by join.