From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6470 Path: news.gmane.org!not-for-mail From: JeanBenabou Newsgroups: gmane.science.mathematics.categories Subject: A well known result Date: Fri, 14 Jan 2011 02:24:18 +0100 Message-ID: <50439B7F-248A-420A-94DD-6ED7D5B11224@wanadoo.fr> Reply-To: JeanBenabou NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1295032210 7358 80.91.229.12 (14 Jan 2011 19:10:10 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 14 Jan 2011 19:10:10 +0000 (UTC) To: "Prof. Peter Johnstone" , Categories , Original-X-From: majordomo@mlist.mta.ca Fri Jan 14 20:10:06 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 1Pdp2E-0001ek-8O for gsmc-categories@m.gmane.org; Fri, 14 Jan 2011 20:10:06 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:54274) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Pdp22-0000hN-TF; Fri, 14 Jan 2011 15:09:55 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Pdp1q-0007ai-Ht for categories-list@mlist.mta.ca; Fri, 14 Jan 2011 15:09:42 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6470 Archived-At: Dear Peter, In one of my mails I mentioned the following result, which I thought to be original: Proposition: Let P: X --> S be a fibration. The functor P is final iff all its fibers are connected From your answer to that mail, dated December 29, I quote: "please don't deceive yourself that this is a new result. It is a (very) special case of the theorem of Street and Walters ("The comprehensive factorization of a functor", Bull. Amer. Math. Soc. 79, 1973) that the pair (final functors, discrete fibrations) forms a factorization structure on Cat. It's true that this result is not stated in the Elephant (why on earth should it be?), but the Street--Walters factorization (for internal categories) is treated in section B2.5." I tried to prove that my proposition was a consequence of the theorem of Street-Walters which you quoted in you mail, but did not succeed. Then I consulted their original paper, hoping to find there more details which would help me to find a proof. Again in vain. I'm quite sure that you're right, and that my inability to get a proof is entirely due to my mathematical limitations. Thus I'd really be very grateful, if you'd give me a proof, or even a sketch of a proof, that my proposition is an easy consequence of the theorem of Street and Walters. Many thanks in advance and best regards, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]