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Coinductive semantics
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Date: -- (:)
From: Hendrik Tews <tews@t...>
Subject: Re: [Caml-list] Coinductive semantics
skaller <> 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.

   >  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.

   > Then you'll see that duality was always
   > considered by all authors on that subject. 
   Hmm .. correct me if I'm wrong, but weren't initial algebras
   studied well before final coalgebras? Perhaps even before
   category theory existed? 

I mean duality was considered by Gumm, Kurz, Hughes, Goldblatt
and all other people that worked on the Co-Birkhoff theorem even
before they started to work on it.