September 22nd, 2011

beartato phd

(no subject)

My mind is being a bit blown by this 2010 paper titled
"Categorifying Computations into Components via Arrows as Profunctors" by Kazuyuki Asada Ichiro Hasuo.

The punchline, as far as I'm concerned, is just the following sentence:
Hughes's notion of "Arrows" are just strong monads in the bicategory of profunctors

(It is, of course, a folk metatheorem in category theory that all the really important theorems in category theory are of the form "Xs are just Ys in the category of Zs", such that at least one of the X, Y, Z is almost too complicated to understand by itself. Which of the X, Y, Z has that property may, however, vary from person to person...)

So by now I know in my guts what a monad is, what it feels like, what it likes for breakfast on a Sunday morning in the springtime, etc. etc. but I'm really fuzzy on what Arrows "really are", even though I know their definition. So this paper gives me hope that I can understand them by and end-run through the notion of profunctor, which I'm really starting to warm up to.