From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8267 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: R A brief survey of cartesian functors Date: Fri, 1 Aug 2014 14:10:34 +0200 Message-ID: <5E8053EE-A599-4E2D-A4D0-B4D937A836FA@wanadoo.fr> References: <0453F381-BF68-4CFC-8FD6-6A3B62D3529D@cs.bham.ac.uk> 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 1406938982 1365 80.91.229.3 (2 Aug 2014 00:23:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 2 Aug 2014 00:23:02 +0000 (UTC) Cc: Categories To: Paul Levy Original-X-From: majordomo@mlist.mta.ca Sat Aug 02 02:22:56 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 1XDN68-0007rn-4K for gsmc-categories@m.gmane.org; Sat, 02 Aug 2014 02:22:56 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49706) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XDN5j-0000nz-9s; Fri, 01 Aug 2014 21:22:31 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XDN5j-0004Oe-QN for categories-list@mlist.mta.ca; Fri, 01 Aug 2014 21:22:31 -0300 In-Reply-To: <0453F381-BF68-4CFC-8FD6-6A3B62D3529D@cs.bham.ac.uk> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8267 Archived-At: Dear Paul, I have received many mails concerning the posting you mentioned and I = try to answer all of them with precision. This takes a lot of time. I = shall answer yours, in detail, about the general definition of cartesian = functors in terms of disdtributors very soon. But the precise answer = shall be a bit long, so please be patient.=20 All I can say now is that, if P: X --> S is a prefoliation the general = definition reduces to the simple one I have given (the proof is not = trivial). Moreover some of the results of the theorem require the assumption that = P is a prefoliation. Hardly surprising, if you want to prove stronger = results you need stronger assumptions. The ones I make are, in a sense, = minimal. You'll have I think noticed that for 7) I need P to be a = foliation. Thanks for your interest. Best regards, Jean Le 1 ao=FBt 2014 =E0 12:35, Paul Levy a =E9crit : >=20 > On 28 Jul 2014, at 10:54, Jean B=E9nabou wrote: >=20 >> 2) CARTESIAN FUNCTORS >> Let P: X --> S, P': X' --> S and F: X --> X' be functors such = that P =3D P'F. For every object s of S ,I denote by F_s : X_s --> = X'_s the functor induced by F on the fibers. >> I have a general definition of F being cartesian, without any = assumption on P and P' and without any reference to cartesian maps, but = it uses distributors in an essential manner. >=20 > Please tell us your general definition using distributors. >=20 > Do any of the results in your Theorem 2.3 hold in this more general = setting? >=20 > Paul >=20 >> 2.3. THEOREM. If P is a pre foliation, P' arbitrary, and F is = cartesian, then: >> (1) F is faithful every F_s is. >> (2) F is full iff every F_s is. >> (3) F is essentially surjective iff every F_s is. >> (4) F is final iff every F_s is. >> (5) F is flat iff every F_s is. >> (6) F has a left adjoint iff every F_s has. >> If moreover P is a foliation, then >> (7) F is conservative iff every F_s is. >=20 > -- > Paul Blain Levy > School of Computer Science, University of Birmingham > +44 121 414 4792 > http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]