From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8218 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 13:30:22 +0100 Message-ID: References: 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 1406054051 16439 80.91.229.3 (22 Jul 2014 18:34:11 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 22 Jul 2014 18:34:11 +0000 (UTC) To: =?utf-8?Q?Jean_B=C3=A9nabou?= , Categories Original-X-From: majordomo@mlist.mta.ca Tue Jul 22 20:34:05 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 1X9et2-00030j-Al for gsmc-categories@m.gmane.org; Tue, 22 Jul 2014 20:34:04 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47641) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X9esQ-0002Nr-8J; Tue, 22 Jul 2014 15:33:26 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X9esP-0002w9-Mu for categories-list@mlist.mta.ca; Tue, 22 Jul 2014 15:33:25 -0300 In-Reply-To: X-Mailer: iPad Mail (11D257) Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8218 Archived-At: Dear Jean, Thank you for your detailed comments. Something I should say straight away is that the duality argument I had in m= ind, dualizing 2-cells, might be OK to deal with left adjoints to reindexing= but was completely wrong for right adjoints. Already, Richard Garner and Cl= audio Ermida (thanks to both of them) have shown me that it doesn't do the j= ob. I also want to stress that at no point did I intend to set up my own definit= ion 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 pseudoa= lgebra 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 so= mewhere.=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." I agree that the concept I have been using includes cleavage (and, for a bif= ibration, cocleavage). I cannot assume AC in what I do, and I rather imagine= d that structure something like the Chevalley criterion was needed in order t= o deal with its absence. However, I admit I am not so familiar with the full= y general notion of fibration. For me the Chevalley condition seemed enough t= o do what I needed in the 2-category Loc of locales and my remarks were base= d on that experience. Best wishes, Steve. > On 20 Jul 2014, at 17:18, Jean B=C3=A9nabou wrot= e: >=20 > A few weeks ago there has been a discussion about stability by composition= of fibrations, bifibrations, and similar notions. Obviously the results dep= end on how such notions are defined. I would like to make a few comments, i= n particular about Steve Vickers' mail, since all the other participants to t= he discussion seemed to accept his approach. >=20 > (I ) VICKERS' DEFINITION OF FIBRATION > if C is a 2-category with comma objects and 2-pullbacks, a one cell p: B -= > A is a fibration iff it satisfies the Chevalley condition. >=20 > Let us test this definition in special cases. > If S is a category with finite limits the 2-category Cat(S) of internal c= ategories in S satisfies Vickers' conditions hence we know when an internal f= unctor is a fibration.=20 > Let Set be the category of sets, except WE DON'T ASSUME THE AXIOM OF CHOIC= E (AC). Then Cat(Set), abbreviated by Cat, is the 2-Category of small catego= ries. > An easy verification shows that a functor p: B -> A satisfies the Chevall= ey condition iff it is a fibration which admits a cleavage. Thus Vickers' ar= gument, in that case, gives as result: fibrations WHICH ADMIT A CLEAVAGE are= stable by composition.=20 > On the other hand, it is easy to show that: Every small fibration has a cl= eavage is equivalent to AC. This well known fact can be very much strengthen= ed by the following example: >=20 > If AC does not hold in Set, one can construct in Cat a bifibration p: B -= > A with internal products and coproducts where A and B are pre-ordered set= s, with pullbacks preserved by p, every map of B is both cartesian and cocar= tesian, and add each of the following conditions: > (i) p has neither a cleavage nor a cocleavage.=20 > (ii) A bit surprisingly: p is a split fibration but has no cocleavage. > (iii) Dual of (ii): p is a cosplit cofibration but has no cleavage. >=20 > And of course we don't need AC to show that arbitrary fibrations in Cat ar= e stable by composition. >=20 ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]