From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8266 Path: news.gmane.org!not-for-mail From: Paul Levy Newsgroups: gmane.science.mathematics.categories Subject: Re: A brief survey of cartesian functors Date: Fri, 1 Aug 2014 11:35:58 +0100 Message-ID: References: Reply-To: Paul Levy NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=ISO-8859-1; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1406938870 32102 80.91.229.3 (2 Aug 2014 00:21:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 2 Aug 2014 00:21:10 +0000 (UTC) Cc: Categories To: =?ISO-8859-1?Q?Jean_B=E9nabou?= Original-X-From: majordomo@mlist.mta.ca Sat Aug 02 02:21:05 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 1XDN4K-0005bR-QO for gsmc-categories@m.gmane.org; Sat, 02 Aug 2014 02:21:04 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49700) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XDN3z-0000aM-8W; Fri, 01 Aug 2014 21:20:43 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XDN3z-0004Mu-Mm for categories-list@mlist.mta.ca; Fri, 01 Aug 2014 21:20:43 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8266 Archived-At: On 28 Jul 2014, at 10:54, Jean B=E9nabou wrote: > 2) CARTESIAN FUNCTORS > Let P: X --> S, P': X' --> S and F: X --> X' be functors such =20 > that P =3D P'F. For every object s of S ,I denote by F_s : X_s --> =20= > X'_s the functor induced by F on the fibers. > I have a general definition of F being cartesian, without any =20 > assumption on P and P' and without any reference to cartesian maps, =20= > but it uses distributors in an essential manner. Please tell us your general definition using distributors. Do any of the results in your Theorem 2.3 hold in this more general =20 setting? Paul > 2.3. THEOREM. If P is a pre foliation, P' arbitrary, and F is =20 > cartesian, then: > (1) F is faithful iff 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. -- 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/ ]