Arrows (was Re: Proposal: Applicative => Monad: Call for consensus)

[Ross Paterson]

On Wed, Jan 05, 2011 at 12:10:36PM -0500, Tyson Whitehead wrote:
> I wonder if Patai hit upon the root of the issue when in his blog (which 
> includes a nice hierarchy diagram) he suggested/proposed that "Arrow is 
> strictly the intersection of Applicative and Category".
> 
> http://just-bottom.blogspot.com/2010/04/programming-with-effects-story-so-far.html

I don't think that's quite true.  Category(~>) + forall a. Applicative(a~>)
gives you the arrow constants, but not the arrow laws, without some additional
conditions relating the two, like

	fmap f (g >>> h) = g >>> fmap f h
	liftA2 (,) id id = arr (\x -> (x,x))
	liftA2 const f id = f
	arr f >>> g &&& h = (arr f >>> g) &&& (arr f >>> h)
	first f >>> first g = first (f >>> g)