On 1 Aug 2007, at 21:23, 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. 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 : ???
I'm not sure that's the right question. A monoid is a category with one object. A group is a category with one object, where every arrow is an iso. A groupoid is a category (with potentially more than one object), where every arrow is an iso. A monad is monoid in the category of endofunctors. Your "groupad", i.e. a group in the category of endofunctors, would boil down to having a monad m that has an unreturn :: m a -> a. In the light of things like unsafePerformIO, this is maybe not what you want. More interesting, however, is generalizing a monad to a monoid in functor categories in general, as opposed to just endofunctors. You could call this a "monadoid". I've heard Tarmo Uustalu talk about this once (Kan-extensions, coends, *mumble mumble mumble*). Mike Spivey's MSFP paper (http://spivey.oriel.ox.ac.uk/mike/msfp.pdf) mentions A-monads, which is a step in that direction. Anyhow, just a thought. Wouter