From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8250 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: Mon, 28 Jul 2014 19:36:57 +0200 Message-ID: References: <8C57894C7413F04A98DDF5629FEC90B1DB632C@Pli.gst.uqam.ca> 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 1406682107 28149 80.91.229.3 (30 Jul 2014 01:01:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 30 Jul 2014 01:01:47 +0000 (UTC) Cc: Categories To: =?iso-8859-1?Q?=22Joyal=2C_Andr=E9=22?= Original-X-From: majordomo@mlist.mta.ca Wed Jul 30 03:01:41 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 1XCIGy-00057P-QB for gsmc-categories@m.gmane.org; Wed, 30 Jul 2014 03:01:40 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49061) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XCIGc-0004QE-FL; Tue, 29 Jul 2014 22:01:18 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XCIGc-0007pC-Vi for categories-list@mlist.mta.ca; Tue, 29 Jul 2014 22:01:18 -0300 In-Reply-To: <8C57894C7413F04A98DDF5629FEC90B1DB632C@Pli.gst.uqam.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8250 Archived-At: Dear Andr=E9 Your guess is quite correct. More generally every full and faithful = functor is a foliation.=20 Let me call a functor P: X --> S locally full and faithful (lff) iff = for each object x of X the obvious functor X/x --> S/P(x) is full and = faithful. Such functors are also characterized by : Every map of X is = hypercartesian. Thus they are foliatiions. Now every full and faithful = functor is lff. Hence is a foliation. I mentioned in my mail that there are many foliations which are not = fibrations, this is a typical example. It shows how much more general = than (pre) fibrations (pre) foliations can be. To give an easy but important application, let me note that, if X is a = groupoid avery functor P: X --> S, where S is arbitrary, is a = foliation, because all the maps of X are isos, hence hypercartesian. This gives me the opportunity to explain condition (ii) for cartesian = functors in a special case. Suppose P: X --> S, P' : X' --> S and F; X --> X' verify P =3D P'F, where = X, X' and S are groups. Then P and P' are foliations and F preserves = cartesian maps. However F need not be cartesian. More precisely F = satisfies (ii) iff P and P' have same image in S. In that case the = theorem says: F is a mono (resp an epi) iff its restriction Ker(P) = --> Ker(P') is a mono (resp an epi). This would be obviously false = without (ii)=20 To complete the picture let us see that (i) =3D> (ii) when P is a = fibration. In that case P is surjective, i.e. Im(P) =3D S contains = Im(P') . But P =3D P'F =3D> Im(P) is contained in Im(P'), hence the = equality required. Thank you for having given me the occasion to explicit some examples, an = in particular to show that (ii) is meaningful. Bien amicalement, Jean Le 28 juil. 2014 =E0 17:53, Joyal, Andr=E9 a =E9crit : > Dear Jean, >=20 > I apologise for my ignorance of your work. >=20 > I guess that an equivalence of categories P:X-->S is always a = foliation, but not > a fibration, unless it is surjective on objects. >=20 > -Andr=E9 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]