Proposal: Add Compositor class as superclass of Arrow
apfelmus at quantentunnel.de
Fri Oct 19 05:01:42 EDT 2007
Cale Gibbard wrote:
> fmap (f . g) x = fmap f (fmap g x)
> (f . g) . x = f . (g . x)
> fmap id x = x
> id . x = x
Nice! Can this be done in Category Theory, too? I mean, it would be nice
to internalize morphism, functors, natural transformations, ... in one
and the same category (like Hask), so there's less fuss. I.e. given a
category C , construct an category C\infty that is basically the same
as C but also contains the functors, natural transformations etc. of
C and has this handy infix (.) operation.
> I've tried this out for a while, and it is actually rather nice to use
> in many cases. Functor application is common enough that having a
> one-character representation for it is great.
I can't remember using fmap/liftM very often, but if I use `liftM`, then
often in infix notation, so and infix symbol for fmap is indeed a very
However, (.) in that role confuses me because I always think that the
right argument should be function. In other words, I'm fine with
print . sum . map read . lines . readFile
( with a hypothetical instance Category (a -> IO b) ) while your
proposal gives rise to
show . sum . map read . lines . readFile "foo.txt"
which makes me feel ill. In my opinion, function composition and
function application should have separate notations. The new (.) blurs
these lines too much for my taste (i.e. (.) :: (a -> b) -> Id a -> Id b)
and I prefer <$> (or even plain $) for fmap .
In addition, I always longed for categories without an embedding (a ->
b) -> c a b , they keep popping up while I program in Haskell and more
often than I need infix fmap . Also, I dislike (>>>) or (<<<) and very
much prefer (.) for them.
But in the end, we can have both worlds of (.) without name clash!
Simply annotate functors with the category they operate on :)
class Category c => Functor c f where
(.) :: c a b -> f a -> f b
instance Functor (->)  where
(.) = map
instance Category c => Functor c (c d) where
(.) = `o`
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