From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8222 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Composition of Fibrations Date: Tue, 22 Jul 2014 06:24:32 +0200 Message-ID: References: <3E52EFB7-7955-47B1-9B00-9F6F6152BBC1@cs.bham.ac.uk> <32AB43B0-58DA-4375-A4FD-6C84F4E527EA@wanadoo.fr> <6EFFC44F-E933-412B-89F2-C33B598D78B0@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=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1406054284 19471 80.91.229.3 (22 Jul 2014 18:38:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 22 Jul 2014 18:38:04 +0000 (UTC) Cc: Categories To: Steve Vickers Original-X-From: majordomo@mlist.mta.ca Tue Jul 22 20:37:58 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 1X9ewm-0006Ar-W8 for gsmc-categories@m.gmane.org; Tue, 22 Jul 2014 20:37:57 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47669) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X9ewS-00035Q-Bf; Tue, 22 Jul 2014 15:37:36 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X9ewR-00033i-QM for categories-list@mlist.mta.ca; Tue, 22 Jul 2014 15:37:35 -0300 In-Reply-To: <6EFFC44F-E933-412B-89F2-C33B598D78B0@cs.bham.ac.uk> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8222 Archived-At: Dear Steve, At least one ambiguity is solved. Chevalley gave as criterium (for = opfibrations) that the arrow=20 p~ : =CE=A6E -> p/B in your mail has a left adjoint with unit the = identity. When the 2-category is Cat this condition is satisfied iff p is an = opfibration which has an opcleavage. The choice of the adjoint defines = the opcleavage. Let us for the sake of precision call Street criterium the existence of = a left adjoint with unit an iso, and Street opfibrations (in Cat) the = functors which satisfy this condition.=20 They need not be opfibrations in the sense of Grothendieck which is = almost unanimously adopted. It is unfortunate to have given them the = name of (op)fibrations, not only because of the ambiguity as we have = seen, but because the fibers are meaningless, in particular the fibers = over two isomorphic objects of the base B need not be isomorphic.=20 I'm almost sure that Neil Ghani, Richard Garner, Claudio Hermida and = Thomas Streicher meant Grothendieck fibrations, and the genuine = Chevalley condition in your answer, as I did. Regards, Jean =20 > Dear Jean, >=20 > Street's result is as follows. The arrow p: E -> B is a 0-fibration = over B if and only if the arrow > p~ : =CE=A6E -> p/B > corresponding to the 2-cell >=20 > =CE=A6E --pd1--> B > | || > d0 || > | p=CE=BB =3D> || > v || > E --p------> B >=20 > has a left adjoint with unit an isomorphism. >=20 > Here =CE=A6E =3D E/E and p/B are comma objects, d0 and d1 are = projections, and =CE=BB is the canonical 2-cell in a comma square (in = this case for =CE=A6E). 0-fibration is opfibration. >=20 > Regards, >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]