From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6457 Path: news.gmane.org!not-for-mail From: JeanBenabou Newsgroups: gmane.science.mathematics.categories Subject: Fibrations in a 2-category Date: Tue, 11 Jan 2011 08:31:54 +0100 Message-ID: Reply-To: JeanBenabou NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1294749612 2345 80.91.229.12 (11 Jan 2011 12:40:12 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 11 Jan 2011 12:40:12 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Tue Jan 11 13:40:08 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 1PcdW7-00013Q-CI for gsmc-categories@m.gmane.org; Tue, 11 Jan 2011 13:40:03 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:44238) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PcdVy-0000WU-7g; Tue, 11 Jan 2011 08:39:54 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PcdVr-0004A6-7N for categories-list@mlist.mta.ca; Tue, 11 Jan 2011 08:39:47 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6457 Archived-At: I have seen very often the following "abstract" definition of a =20 fibration in a 2-category C : A map (i.e. a 1-cell) p: X --> S is a fibration iff for each object Y =20= of C the functor C(Y,p): C(Y,X) --> C(Y,S) is a fibration (in the =20 usual sense) which depends "2-functorially" on Y. Such an "obvious" definition is much too naive and does not give the =20 correct notion in most examples. 1- Even if C=3D Cat, the 2-category of (small) categories, a fibration =20= in the abstract sense is a Grothendieck fibration which admits a =20 cleavage. Thus if we don't assume AC, which we don't need to define =20 fibrations, it does not coincide with the usual one. 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. Best to all, Jean =20= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]