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