From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8220 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Composition of Fibrations Date: Mon, 21 Jul 2014 21:06:29 +0100 Message-ID: References: <3E52EFB7-7955-47B1-9B00-9F6F6152BBC1@cs.bham.ac.uk> <32AB43B0-58DA-4375-A4FD-6C84F4E527EA@wanadoo.fr> Reply-To: Steve Vickers NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1406054194 18474 80.91.229.3 (22 Jul 2014 18:36:34 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 22 Jul 2014 18:36:34 +0000 (UTC) Cc: Categories To: =?utf-8?Q?Jean_B=C3=A9nabou?= Original-X-From: majordomo@mlist.mta.ca Tue Jul 22 20:36:26 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 1X9evI-0004uq-Uy for gsmc-categories@m.gmane.org; Tue, 22 Jul 2014 20:36:25 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47655) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X9ev1-0002mr-01; Tue, 22 Jul 2014 15:36:07 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X9euz-00030B-S7 for categories-list@mlist.mta.ca; Tue, 22 Jul 2014 15:36:05 -0300 In-Reply-To: <32AB43B0-58DA-4375-A4FD-6C84F4E527EA@wanadoo.fr> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8220 Archived-At: Dear Jean, Street's result is as follows. The arrow p: E -> B is a 0-fibration over B i= f and only if the arrow p~ : =CE=A6E -> p/B corresponding to the 2-cell =CE=A6E --pd1--> B | || d0 || | p=CE=BB =3D> || v || E --p------> B has a left adjoint with unit an isomorphism. Here =CE=A6E =3D E/E and p/B are comma objects, d0 and d1 are projections, a= nd =CE=BB is the canonical 2-cell in a comma square (in this case for =CE=A6= E). 0-fibration is opfibration. Regards, Steve. > On 21 Jul 2014, at 19:02, Jean B=C3=A9nabou wrot= e: >=20 > Dear Steve, >=20 > Thank you for your prompt answer.=20 >=20 > Let me first clarify a possible ambiguity.=20 > The Street fibrations I was referring to are defined in his paper:=20 > Fibrations in bicategories. Cahiers Top. Geom. Diff. 21 (1980) > When the bicategory is Cat, they do not coincide with the usual fibrations= . In particular every equivalence is a Street fibration. >=20 > There might be another ambiguity about what you call the Chevalley criteri= um. Could you please tell me with precision what it is (I assume p: B -> A i= s a map in a 2-category C with comma objects and 2-pullbacks) >=20 > I shall come back to mathematical questions as soon as these two ambiguiti= es are solved. >=20 > Best wishes, >=20 > Jean >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]