On 26 Jan 2005, at 08:41, Keean Schupke wrote:
I cannot find any reference to MonadPlus in category theory. At a guess I would say that it was the same as a Monad except the operators are id and co-product (or sum)... That would mean the 'laws' would be exactly the same as a Monad, just with (0,+) instead of (1,*)...
I don't know if the categorists have a word for it. It isn't "the same as a Monad except", it's rather "the same as a Monad and also...". Note that translating between the normal categorical point of view and the haskell one is something of a pain since programmers think with Kleisli arrows (computations). A MonadPlus is a Monad T with, in addition, the structure of a monoid on all types T a, satisfying the some conditions. I follow Wadler and use zero and ++ for the monoid. Unit conditions: If you apply a monadic computation (that is, Kleisli arrow) to the zero object you get zero, that there is a special zero computation (that is concretely \x -> zero) which gives you the zero object when you apply it to anything. Compatibility of ++ with computations: The first condition just says that if you apply a computation to the sum of two objects, that should be the same as applying the computation twice, once to each object, and adding the results. (m ++ n) >>= f == (m >>= f) ++ (n >>= f) The two slightly more subtle conditions observe that it is possible to 'lift' the monoid operation to Kleisli arrows, i.e. if you can add objects, then you can add computations (of the same type): (+++) : (a -> m b) -> (a -> m b) -> (a -> m b) \f g x -> f x ++ g x Then you require that if you first add two computations and apply them to a single object, you get the same result as applying the computations separately and adding the results.. m >>= (f +++ g) == (m >>= f) ++ (m >>= g) ----- If these are a common categorical notion then I'd expect them to be called 'additive monads' or something, but I can't immediately track down a reference. If you translate the notion back to the more standard categorical approach, using mu (join) and eta (return), it may be enlightening, but off hand I can't quite see what the rules look like. Jules