Math alert: mild category theory. Greg Meredith wrote:
> But, along these lines i have been wondering for a while... the monad laws > present an alternative categorification of monoid. At least it's > alternative to monoidoid. I wouldn't call monads categorifications of monoids, strictly speaking. A monad is a monoid object in a category of endofunctors (which is a monoidal category under composition). What do you mean by a 'monoidoid'? I only know it as a perverse synonym of 'category' :-). > In the spirit of this thought, does anyone know of an > expansion of the monad axioms to include an inverse action? Here, i am > following an analogy > > monoidoid : monad :: groupoid : ??? First of all, I don't actually know the answer. The canonical option would be a group object in the endofunctor category (let's call the latter C). This does not make sense, however: in order to formulate the axiom for the inverse, we would need the monoidal structure of C (composition of functors) to behave more like a categorical product (to wit, it should have diagonal morphisms diag :: m a -> m (m a) ). Maybe there is a way to get it to work, though. After all, what we (in FP) call a commutative monad, is not commutative in the usual mathematical sense (again, C does not have enough structure to even talk about commutativity). > My intuition tells me this could be quite generally useful to computing in > situation where boxing and updating have natural (or yet to be discovered) > candidates for undo operations. i'm given to understand reversible > computing > might be a good thing to be thinking about if QC ever gets real... ;-) If this structure is to be grouplike, the inverse of an action should be not only a post-inverse, but also a pre-inverse. Is that would you have in mind? (If I'm not making sense, please shout (or ignore ;-) ).) Greetings, Arie _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe