[Haskell-cafe] Composition Operator

[Dan Weston]

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