From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8219 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: Mon, 21 Jul 2014 20:02:28 +0200 Message-ID: References: <3E52EFB7-7955-47B1-9B00-9F6F6152BBC1@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 1406054102 17257 80.91.229.3 (22 Jul 2014 18:35:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 22 Jul 2014 18:35:02 +0000 (UTC) Cc: Categories To: Steve Vickers Original-X-From: majordomo@mlist.mta.ca Tue Jul 22 20:34:56 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 1X9etr-0003hu-RI for gsmc-categories@m.gmane.org; Tue, 22 Jul 2014 20:34:56 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47648) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X9etZ-0002XW-94; Tue, 22 Jul 2014 15:34:37 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X9etY-0002xe-Pt for categories-list@mlist.mta.ca; Tue, 22 Jul 2014 15:34:36 -0300 In-Reply-To: <3E52EFB7-7955-47B1-9B00-9F6F6152BBC1@cs.bham.ac.uk> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8219 Archived-At: Dear Steve, Thank you for your prompt answer.=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. There might be another ambiguity about what you call the Chevalley = criterium. Could you please tell me with precision what it is (I assume = p: B -> A is a map in a 2-category C with comma objects and 2-pullbacks) I shall come back to mathematical questions as soon as these two = ambiguities are solved. Best wishes, Jean Le 21 .. 2014 =E0 14:30, Steve Vickers a =E9crit : > Dear Jean, >=20 > Thank you for your detailed comments. >=20 > Something I should say straight away is that the duality argument I = had in mind, dualizing 2-cells, might be OK to deal with left adjoints = to reindexing but was completely wrong for right adjoints. Already, = Richard Garner and Claudio Ermida (thanks to both of them) have shown me = that it doesn't do the job. >=20 > I also want to stress that at no point did I intend to set up my own = definition of fibration. I was following Street's "Fibrations and = Yoneda's lemma in a 2-category", which defines fibrations as those = 1-cells that carry pseudoalgebra structure for a certain 2-monad, and = then proves (Proposition 9) that this is equivalent to what Street = refers to as the Chevalley condition. If "Vickers' definition" is not = equivalent to that then I have made a mistake somewhere.=20 >=20 > Have you found a discrepancy between the "Vickers definition" and = Street? At one point you write "Of course, I don't refer here to = Street's notion which describes a totally different kind of fibration, = stable by equivalences." >=20 > I agree that the concept I have been using includes cleavage (and, for = a bifibration, cocleavage). I cannot assume AC in what I do, and I = rather imagined that structure something like the Chevalley criterion = was needed in order to deal with its absence. However, I admit I am not = so familiar with the fully general notion of fibration. For me the = Chevalley condition seemed enough to do what I needed in the 2-category = Loc of locales and my remarks were based on that experience. >=20 > Best wishes, >=20 > Steve. >=20 >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]