[Haskell-cafe] Composition Operator
Dan Weston
westondan at imageworks.com
Mon Sep 24 22:11:09 EDT 2007
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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