Proposal: Add Compositor class as superclass of Arrow
apfelmus
apfelmus at quantentunnel.de
Thu Oct 25 12:03:22 EDT 2007
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
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?
Regards,
apfelmus
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