From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3986 Path: news.gmane.org!not-for-mail From: Lutz Schroeder Newsgroups: gmane.science.mathematics.categories Subject: Re: locally cartesian closed categories Date: Tue, 09 Oct 2007 09:34:19 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019644 11195 80.91.229.2 (29 Apr 2009 15:40:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:44 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Oct 9 22:00:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 22:00:10 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfPmY-0000yj-9P for categories-list@mta.ca; Tue, 09 Oct 2007 21:50:38 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 43 Original-Lines: 57 Xref: news.gmane.org gmane.science.mathematics.categories:3986 Archived-At: > I agree with Jean Benabou, Fred Linton and Vaughan Pratt that the > definition of a locally cartesian closed category should NOT require > a terminal object. =20 [...] > I confess that I'm a bit surprised to find that the consensus agrees > with me, so to set matters straight I should also point out that my > argument applies equally to elementary toposes and other familiar > structures of categorical logic. Such as cartesian closed categories, for instance. I would like to take the opportunity to point to my paper "Life without the terminal type" in CSL 2001, where I prove that every "almost" cartesian category, i.e. one without a terminal object, extends uniquely to a cartesian closed category with terminal object. There is also a similar result for toposes; the wording is not quite as straightforward as for cartesian closed categories, as one has to formulate (say) the definition of a subobject classifier without reference to a global element True. I recall having thought about locally cartesian closed categories as well, but I do not think I really got anywhere (and actually I just see there's a remark in the paper that says as much). Lutz Schr=F6der --=20 ------------------------------------------------------------------ PD Dr. Lutz Schr=F6der office @ Universit=E4t Bremen: Senior Researcher Cartesium 2.051 Safe and Secure Cognitive Systems Enrique-Schmidt-Str. 5 DFKI-Lab Bremen FB3 Mathematik - Informatik Robert-Hooke-Str. 5 Universit=E4t Bremen D-28359 Bremen P.O. Box 330 440 D-28334 Bremen phone: (+49) 421-218-64216 Fax: (+49) 421-218-9864216 mail: Lutz.Schroeder@dfki,de www.dfki.de/sks/staff/lschrode ------------------------------------------------------------------ ------------------------------------------------------------- Deutsches Forschungszentrum f=FCr K=FCnstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Gesch=E4ftsf=FChrung: Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) Dr. Walter Olthoff Vorsitzender des Aufsichtsrats: Prof. Dr. h.c. Hans A. Aukes Amtsgericht Kaiserslautern, HRB 2313 -------------------------------------------------------------