On Wed, 2006-01-18 at 13:58 +0100, Hendrik Tews wrote:
> skaller <skaller@users.sourceforge.net> writes:
> 
>    > Nobody is interested in final coalgebras in Set^op. 
>    
>    Why not? This is really the key point of misunderstanding
>    I think. I'm not disputing your claim, I'm asking why not?
>    Perhaps they should be?
>    
> Coalgebras in Set^op are for all intents and purposes identical
> to algebras in Set. If you want to study them, study them as
> algebras in Set. You will see nothing new if you look at these
> objects as coalgebras in Set^op. That's what duality means.
> 
> Looking at an object through a mirror you see precisely what you
> can see looking at the object itself.

Perhaps my analysis is naive. But consider a simpler case
of products and sums. They're dual concepts, are they not?

In Ocaml we have representations of both, each can be used
with reasonable utility -- there is a degree of symmetry,
associated with the duality. It feels good!

Contrast to C, which has products, but the union construction
isn't a sum. And the many other 'popular' languages with 
this weakness.

Sometimes it seems looking in the mirror is good. 
It's what we want. We don't want something new!

>    >  Go out, read the papers on
>    > the Co-Birkhoff theorem! 
>    
>    That's a pretty big ask of someone who isn't a
>    category theorist isn't it? Most mathematicians
>    can't understand category theory .. and I'm just
>    an ordinary programmer :)
>    
> Well, you could try. I guess, that already the introductions
> contain enough information for what you are interested in: the
> duality of the Birkhoff and the Co-Birkhoff theorem. In any case,
> if you don't even try, your speculations about the contents of
> these papers remain wild guesses.

I often do try.. but seemed like a good idea to read Adameck first:

http://www.tac.mta.ca/tac/volumes/14/8/14-08abs.html

Still this is quite heavy going for me.

Incidentally .. if you look in Wikipedia for 'coalgebra' you may
be a bit disappointed.

http://en.wikipedia.org/wiki/Coalgebra


-- 
John Skaller <skaller at users dot sf dot net>
Felix, successor to C++: http://felix.sf.net