From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6474 Path: news.gmane.org!not-for-mail From: JeanBenabou Newsgroups: gmane.science.mathematics.categories Subject: Re: A well known result Date: Fri, 14 Jan 2011 20:56:05 +0100 Message-ID: References: <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=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1295188708 10600 80.91.229.12 (16 Jan 2011 14:38:28 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 16 Jan 2011 14:38:28 +0000 (UTC) To: "Prof. Peter Johnstone" , Categories Original-X-From: majordomo@mlist.mta.ca Sun Jan 16 15:38:22 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 1PeTkM-0003vY-G3 for gsmc-categories@m.gmane.org; Sun, 16 Jan 2011 15:38:22 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:38493) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PeTjy-0000IU-Q2; Sun, 16 Jan 2011 10:37:58 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PeTjw-000590-EG for categories-list@mlist.mta.ca; Sun, 16 Jan 2011 10:37:56 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6474 Archived-At: Dear Peter, I stand corrected. My proposition is indeed an immediate consequence =20 of Street and Walters. It is also a consequence of much more general =20 results of mine on foliated categories which I didn't mention, and I =20= didn't realize that this special case was easy. This is no excuse. I =20 was careless, and have to "pay" for this carelessness. this is why I =20 make my answer public although your mail was addressed only to me. There are many mathematical questions I asked you, which you didn't =20 answer. I hope this mail will incite you to answer some of them. Best regards, Jean Le 14 janv. 11 =E0 18:54, Prof. Peter Johnstone a =E9crit : > Dear Jean, > > The derivation seems simple enough to me. Street and Walters showed =20= > that > a functor is final iff it is orthogonal to the class of discrete > fibrations. In particular this applies to fibrations which are final > functors; but any fibration admits a factorization through the > discrete fibration whose fibres are the connected components of > the original fibres. Hence, if a fibration is orthogonal to discrete > fibrations, its fibres must be connected. The converse is similar. > > Best regards, > Peter > > On Fri, 14 Jan 2011, JeanBenabou wrote: > >> Dear Peter, >> >> In one of my mails I mentioned the following result, which I =20 >> thought to be original: >> >> Proposition: Let P: X --> S be a fibration. The functor P is final =20= >> iff all its fibers are connected >> >> =46rom your answer to that mail, dated December 29, I quote: >> >> "please don't deceive yourself that this is a new result. It is a =20 >> (very) >> special case of the theorem of Street and Walters ("The comprehensive >> factorization of a functor", Bull. Amer. Math. Soc. 79, 1973) that =20= >> the >> pair (final functors, discrete fibrations) forms a factorization =20 >> structure >> on Cat. It's true that this result is not stated in the Elephant =20 >> (why on >> earth should it be?), but the Street--Walters factorization (for =20 >> internal >> categories) is treated in section B2.5." >> >> I tried to prove that my proposition was a consequence of the =20 >> theorem of Street-Walters which you quoted in you mail, but did =20 >> not succeed. Then I consulted their original paper, hoping to find =20= >> there more details which would help me to find a proof. Again in =20 >> vain. >> >> I'm quite sure that you're right, and that my inability to get a =20 >> proof is entirely due to my mathematical limitations. >> >> Thus I'd really be very grateful, if you'd give me a proof, or =20 >> even a sketch of a proof, that my proposition is an easy =20 >> 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/ ]