Re: [Haskell-cafe] Category theory monad <----> Haskell monad

The explanation given below might be a bit heavy for someone who didn't know much about category theory. For those individuals I'd recommend Phil Wadler's papers: http://homepages.inf.ed.ac.uk/wadler/topics/monads.html I especially recommend "Monads for Functional Programming", "The Essence of Functional Programming" and "Comprehending Monads". Basically, though, the Haskell implementation _is_ the category theoretic definition of monad, with bind/return used instead of (f)map/join/return as described below. Mike

Date: Thu, 18 Aug 2005 20:39:37 -0400 From: Cale Gibbard Cc: haskell-cafe@haskell.org

On 14/08/05, Carl Marks wrote:

...

Is there any text/article which makes precise/rigorous/explicit the connection between the category theoretic definition of monad with the haskell implementation?

Well, a monad over a category C is an endofunctor T on C, together with a pair of natural transformations eta: 1 -> T, and mu: T^2 -> T such that 1) mu . (mu . T) = mu . (T . mu) 2) mu . (T . eta) = mu . (eta . T) = id_C

In Haskell, a monad is an endofunctor on the category of all Haskell types and Haskell functions between them. Application of the endofunctor to an object is given by applying a type constructor (the one which is made an instance of the Monad class). Application of the endofunctor to a function is carried out by fmap or liftM. The natural transformation eta is called return, and mu is called join (found in the Monad library).

Haskell uses a somewhat different (but equivalent) basis for a monad, in that it is not map, return, and join which need defining to make a type an instance of the Monad class, but return and (>>=), called "bind" or "extend".

One can define bind in terms of fmap, and join as x >>= f = join (fmap f x)

and one can get back join and fmap from return and bind: join x = x >>= id fmap f x = x >>= (return . f)

hope this helps, - Cale _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe