Proposal: Add Compositor class as superclass of Arrow
haskell at list.mightyreason.com
haskell at list.mightyreason.com
Thu Oct 25 12:47:35 EDT 2007
apfelmus wrote:
> Conal Elliott wrote:
>> i'm missing a piece of reasoning. how about having &&& as primitive
>> as in
>> your Cartesian proposal, but without the fst/&&& and snd/&&& laws?
>> you could still introduce those laws in a subclass that does not
>> include Arrow.
>
> That would make me feel uncomfortable. At least, I'd drop the name
> "Cartesian" which intends to allude to "cartesian category".
>
> I can't quite put my finger on it, but I think it's the following: since
> &&& is somewhat abstract, the best way to characterize it is by laws. I
> mean, it's the same for monads and return . The fact that return has
> no side-effects is captured in its entirety by the laws
>
> return x >>= f = f x
> m >>= return = m
>
> In other words, return is determined uniquely by those two laws.
> Likewise, the three laws for fst, snd and &&& uniquely determine (up
> to isomorphism) the notion of "cartesian product" in any category (see
> also http://en.wikipedia.org/wiki/Product_%28category_theory%29).
>
> If those laws don't hold, I think the most compelling characterization
> of &&& is indeed in terms of the decomposition
>
> f &&& g = first f . second g
I think that gets the order of side effects wrong. I think you mean:
f &&& g = second g . first f
(As one can see from the original class default in CategoryFanOut with CategoryPair)
>
> and laws for the new primitives first and second like
>
> first f . first g = first (f . g)
> second f . second g = second (f . g)
>
> and others.
>
> fst . first f . dup = f
> snd . second f . dup = f
> snd . first f . dup = fst . second f . dup
> ...
>
> It's already tricky to list them all, how to express the fact that
> first and second pass the other value intact while still performing a
> potential side effect? And what about a minimal (but still complete)
> amount of laws?
Well, the type signature of first does not allow much to be done with the
snd argument except id.
The other tricks are: should dup have no side effects? How to specify?
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