From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8259 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: Re: A brief survey of cartesian functors Date: Wed, 30 Jul 2014 03:05:30 +0200 Message-ID: References: <1B862C69106C4B6A83703605D3E6A693@ACERi3> <54F4E17E-FAD3-43D8-89F2-5B9CF1C098D8@wanadoo.fr> <400AFA411832442388CF05F4B409628D@ACERi3> Reply-To: =?iso-8859-1?Q?Jean_B=E9nabou?= NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1283) Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1406751859 15632 80.91.229.3 (30 Jul 2014 20:24:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 30 Jul 2014 20:24:19 +0000 (UTC) Cc: "Categories" To: "George Janelidze" Original-X-From: majordomo@mlist.mta.ca Wed Jul 30 22:24:13 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XCaQ1-0005z0-8b for gsmc-categories@m.gmane.org; Wed, 30 Jul 2014 22:24:13 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49238) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XCaPj-0000Zu-KK; Wed, 30 Jul 2014 17:23:55 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XCaPj-0002mP-FP for categories-list@mlist.mta.ca; Wed, 30 Jul 2014 17:23:55 -0300 In-Reply-To: <400AFA411832442388CF05F4B409628D@ACERi3> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8259 Archived-At: Dear George, Thank you for your mail. I see that all my mathematical arguments have = not convinced you, and that trying to add more would be useless. I respect your opinion although I totally disagree with it. Best regards, Jean Le 29 juil. 2014 =E0 21:58, George Janelidze a =E9crit : > Dear Jean, >=20 > Thank you for your kind words at the beginning of your message, and I = apologize if what I said about "factorization" and "cartesian" was = unclear. >=20 > I did not mean to say that there is any connection between = factorization systems and (pre foliations + cartesian FUNCTORS). What I = was trying to say, was only that the following two constructions are = essentially the same (up to an isomorphism): >=20 > (a) For a fibration C-->X every morphism f in C factors as f =3D me, = where m is a cartesian ARROW and e is a vertical arrow (with respect to = the given fibration). >=20 > (b) For a semi-left-exact reflection C-->X (in the sense of = Cassidy--Hebert--Kelly) every morphism f in C factors as f =3D me, where = m is in M, e is in E, E is the class of all morphisms inverted by C-->X, = and M is its orthogonal class (M can also be defined as the class of = trivial covering morphisms in the sense of Galois theory). >=20 > I know this might sound trivial to you, but I think it is a = fundamental connection, which should be widely known. And I believe that = instead of >=20 > "indexed categories versus fibrations" >=20 > one should sometimes also consider >=20 > "indexed categories versus fibrations versus semi-left-exact = reflections" (this is why I mentioned a "third approach"). >=20 > Let me also add now: according to Cassidy--Hebert--Kelly, the = factorization mentioned in (b), where E is as in (b), and M is merely = its orthogonal class, also exists under certain assumptions much weaker = than semi-left-exactness. >=20 > But again, I never thought that what you do with pre foliations and = cartesian functors is a similar kind of factorization and/or that it is = contained in the Cassidy--Hebert--Kelly paper! >=20 > And I hope you have never felt from me any disrespect of your opinions = and/or of your beautiful ideas and results. >=20 > Best regards, > George >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]