Oh, duh! The only systematically distinguishable value in every type is
undefined, but I went and excluded that value from my fake Hask* class.
Never mind, I'll stop while I'm behind! :(
Dan Weston wrote:
Well, I did footnote in my first e-mail that:
[1] I used the asterisk in the category name Hask* to exclude undefined
values or partial functions
[Although I think I may have flipped the asterisk convention.]
I see what you mean by const False and const True being two different
arrows, but now I don't know how that reconciles with the Wikipedia
Example 3 of http://en.wikipedia.org/wiki/Initial_object
"In the category of pointed sets (whose objects are non-empty sets
together with a distinguished element; a morphism from (A,a) to (B,b)
being a function f : A ? B with f(a) = b), every singleton is a zero
object [i.e. both initial and final]."
I thought I was being safe by "distinguishing" () as my distinguished
element. Where did I go wrong?
Dan Weston
Stefan O'Rear wrote:
On Mon, Sep 24, 2007 at 06:47:05PM -0700, Dan Weston wrote:
Of course I should have proofread this one more time!
What is a point? A point in Hask* is a type with only a single value
in it, from which all other values can be constructed. Every value x
maps trivially into a function (const x), and when you apply this
function to the (only) value of a point, you get x back. There is a
built-in Haskell type () whose only value [besides undefined] is
also called (), so we might as well take the type () as our point:
Actually, a point is any one object, for Hask* it is any one monotype
(e.g. (), [Int], (Char,Double)). The magic of an *initial* object
(i.e. a type with only one nullary constructor such as () that has
only one (defined) value) is that there is a *unique* function
mapping it to any other type. But that's being greedy, since we don't
need a unique function, just any one function. A forgetful function
like const doesn't care which type its second argument is.
() isn't an initial object.
There are no initial objects in Hask-with-?, since every object admits
at least four arrows to Bool (const True, const False, const undefined,
and undefined).
Stefan
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