> A way to categorify elements of objects in a cartesian closed category
> (such as that that sufficiently restricted Haskell takes place in) are
> to view entities of type A as maps () -> A.Mikael Johansson wrote:
This rather inconveniently clashes with the fact that A and () -> A are
two distinct types in Haskell. A is just the "curried" counterpart to ()
-> A, just as A -> B is the curried counterpart to OneTuple A -> B and A
-> B -> C is the (fully) curried counterpart to (A,B) -> C
I take it by your argument that curried and uncurried functions, being
isomorphic, are represented by the same object in your category?
Dan
On Wed, 7 Feb 2007, Benjamin Franksen wrote:
Udo Stenzel wrote:
Benjamin Franksen wrote:
Udo Stenzel wrote:
Sure, you're right, everything flowing in the same direction is
usually
nicer, and in central Europe, that order is from the left to the
right.
What a shame that the Haskell gods chose to give the arguments to (.)
and ($) the wrong order!
But then application is in the wrong order, too. Do you really want to
write (x f) for f applied to x?
No, doesn't follow.
No? Your words: "everything flowing in the same direction".
Of the two definitions
(f . g) x = g (f x)
vs.
(f . g) x = f (g x)
the first one (your prefered one) turns the natural applicative order
around, while the second one preserves it. Note this is an objective
argument and has nothing to do with how I feel about it.
I would guess that one thing lying at the bottom of this particular
disagreement is whether everything is a function, or whether objects are
some qualified class of their own.
A way to categorify elements of objects in a cartesian closed category
(such as that that sufficiently restricted Haskell takes place in) are
to view entities of type A as maps () -> A. If this is the case, then
with g::A -> B and f:: B -> C, we would want our composition to work the
same way regardless of what we feed into it, and so we have two choices
of notation: either
x . g . f
or
f . g . x
(where I'm using the fact that composition in a category is associative,
as to not write the brackets everywhere)
Now, here it is perfectly obvious why Udo's argument applies, and that
inverting (.) and ($) would lead to (x f) meaning f applied to x. In a
way, this is more in agreement with the actual maps we view, since
x . g . f
corresponds to a composition
() -> A -> B -> C ==> () -> C
However, if elements are to be viewed as something different entirely,
not in any way acquainted with functions, then one could state that
function composition should be by right action, and element application
by left action, so as to mimic the unix shell situation. It seems a bit
unnatural in purely functional languages though, and I believe that
one'd lose important intuition that way around.
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