From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: Received: from concorde.inria.fr (concorde.inria.fr [192.93.2.39]) by yquem.inria.fr (Postfix) with ESMTP id 50557BBCC for ; Wed, 18 Jan 2006 13:58:55 +0100 (CET) Received: from pauillac.inria.fr (pauillac.inria.fr [128.93.11.35]) by concorde.inria.fr (8.13.0/8.13.0) with ESMTP id k0ICwsJq001000 for ; Wed, 18 Jan 2006 13:58:55 +0100 Received: from nez-perce.inria.fr (nez-perce.inria.fr [192.93.2.78]) by pauillac.inria.fr (8.7.6/8.7.3) with ESMTP id NAA14935 for ; Wed, 18 Jan 2006 13:58:54 +0100 (MET) Received: from mail.tcs.inf.tu-dresden.de (tcs01.inf.tu-dresden.de [141.76.75.1]) by nez-perce.inria.fr (8.13.0/8.13.0) with ESMTP id k0ICwrpu009749 for ; Wed, 18 Jan 2006 13:58:54 +0100 Received: from localhost (localhost [127.0.0.1]) by mail.tcs.inf.tu-dresden.de (Sun Java System Messaging Server 6.1 HotFix 0.06 (built Nov 11 2004)) with ESMTP id <0ITA00084HE5W800@mail.tcs.inf.tu-dresden.de> for caml-list@inria.fr; Wed, 18 Jan 2006 13:58:53 +0100 (MET) Received: from tcs01.inf.tu-dresden.de ([127.0.0.1]) by localhost (mail.tcs.inf.tu-dresden.de [127.0.0.1]) (amavisd-new, port 10024) with ESMTP id 15224-07 for ; Wed, 18 Jan 2006 13:58:53 +0100 (MET) Received: from ithif59.inf.tu-dresden.de (ithif59.inf.tu-dresden.de [141.76.75.59]) by mail.tcs.inf.tu-dresden.de (Sun Java System Messaging Server 6.1 HotFix 0.06 (built Nov 11 2004)) with ESMTPS id <0ITA000UEHE5B600@mail.tcs.inf.tu-dresden.de> for caml-list@inria.fr; Wed, 18 Jan 2006 13:58:53 +0100 (MET) Received: from tews by ithif59.inf.tu-dresden.de with local (Exim 4.50) id 1EzCtp-00007U-L5 for caml-list@inria.fr; Wed, 18 Jan 2006 13:58:53 +0100 Date: Wed, 18 Jan 2006 13:58:53 +0100 From: Hendrik Tews Subject: Re: [Caml-list] Coinductive semantics In-reply-to: <1137163342.3681.533.camel@rosella> To: caml-list@inria.fr Message-id: MIME-version: 1.0 Content-type: text/plain; charset=us-ascii Content-transfer-encoding: 7BIT X-Virus-Scanned: amavisd-new at tcs.inf.tu-dresden.de References: <43BD6418.4090407@barettadeit.com> <43BE6CAB.2030503@andrej.com> <43C3963D.5030601@tsc.uc3m.es> <1136981974.8962.100.camel@rosella> <43C51C33.2000206@andrej.com> <1137031853.3681.138.camel@rosella> <43C661AF.2080404@andrej.com> <1137102848.3681.268.camel@rosella> <1137163342.3681.533.camel@rosella> User-Agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.4 X-Miltered: at concorde with ID 43CE3B8E.003 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)! X-Miltered: at nez-perce with ID 43CE3B8D.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)! X-Spam: no; 0.00; hendrik:01 tews:01 tews:01 caml-list:01 coinductive:01 semantics:01 gumm:01 hendrik:01 sourceforge:01 writes:01 tu-dresden:01 theorem:01 theorem:01 precisely:01 tcs:01 X-Spam-Checker-Version: SpamAssassin 3.0.3 (2005-04-27) on yquem.inria.fr X-Spam-Level: X-Spam-Status: No, score=0.0 required=5.0 tests=none autolearn=disabled version=3.0.3 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. Hendrik