# 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

```