From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6460 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: Fibrations in a 2-category Date: Wed, 12 Jan 2011 10:42:03 +1100 Message-ID: References: Reply-To: Ross Street NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=ISO-8859-1; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1294789854 24403 80.91.229.12 (11 Jan 2011 23:50:54 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 11 Jan 2011 23:50:54 +0000 (UTC) Cc: Categories To: JeanBenabou Original-X-From: majordomo@mlist.mta.ca Wed Jan 12 00:50:47 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PcnzD-0003T2-Jo for gsmc-categories@m.gmane.org; Wed, 12 Jan 2011 00:50:47 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42689) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Pcnyz-0008TA-93; Tue, 11 Jan 2011 19:50:33 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Pcnyp-00044F-8f for categories-list@mlist.mta.ca; Tue, 11 Jan 2011 19:50:23 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6460 Archived-At: Dear Jean On 11/01/2011, at 6:31 PM, JeanBenabou wrote: > 2- The situation is much worse in more general cases. Suppose E is a =20= > topos (this assumption is much too strong), and take C =3D Cat(E), the = =20 > category of internal categories in E. On can define internal =20 > fibrations, and "fibrations" in the previous "abstract" sense. They =20= > do not coincide. > It all boils down to the following remark: E and (E=B0, Set) are =20 > Toposes, the Yoneda functor E --> (E*,Set) preserves an reflects =20 > limits, but "nothing else" of the internal logic, which is needed to =20= > define internal fibrations. I totally agree. An internal fibration between groups in a topos E is =20= a group morphism whose underlying morphism in E is an epimorphism; for =20= a representable fibration, it is a split epimorphism in E. Jack Duskin =20= alerted me to this many years ago. Never-the-less, the representable notion has had some uses. Actually, =20= Dominic Verity and I also used representably Giraud-Conduch=E9 morphisms = =20 in The comprehensive factorization and torsors, Theory and = Applications =20 of Categories 23(3) (2010) 42-75; whereas there is an internal version (more generally applicable in the =20= way you explain) of these too (in a topos, for example). Have you written or published anything on these internal notions? Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]