[Haskell-cafe] Join and it's relation to >>= and return

Ron de Bruijn rondebruijn at yahoo.com
Mon Jun 7 15:19:36 EDT 2004

Hello,

The last 3 or 4 days I have been studying a lot of
Category Theory so that I would be able to explain the
concept of a monad to some people at the university in
a "learn to present something"-course.

newtype S a = State -> (a,State) -- functor T to map
objects
mapS::(a-> b) -> (S a -> S b)  -- functor T to map
morphisms
unitS :: a -> S a  --\eta
joinS::S(S a)-> S a -- \mu

This is a complete monad using a direct mapping from
Category Theory. I really like it, because it's
mathematically grounded. But I don't know how to map
"return", as I explain below.

Although, I do believe I understand most parts of
it(as in Functor, Natural Transformation and so on),
I don't really understand how the join function
(conventional monadic multiplication as it's called in
the documentation) relates to the >>= and return
functions that are mostly used. I read somewhere at
Google that join can be expresed in terms of >>= and
return, but I couldn't find it anymore.

I also read that the Kleisli triple has more
resemblance to how monads are used in practice, but it
has some additional constraints. I don't understand
why there are these additional constraints and I don't
understand it's relevance, because according to some
book there's a one-one mapping from monads to Kleisli
triples and reverse.

I also don't understand why join is a
"multiplication", because a multiplication takes two
operands (as in 5*4), at least all multiplication
operators I know do so. I think what's happening is
that one structure is taken out of the other. And
then, the two are "merged" in some way. So this way,
there are two operands. Is this correct? (I really
could use a function definition in Haskell of join in
terms of bind and return).

Can somebody explain me (with a lot of detail)  how
this works?

Regards,
Ron

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