[Haskell-cafe] Composition Operator

[Dan Weston]

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
> 
>