From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8270 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: Re: cleavages and choice Date: Fri, 1 Aug 2014 19:52:53 +0200 Message-ID: References: <20140730150643.GC19613@mathematik.tu-darmstadt.de> <89048344-26F3-448F-8B41-9FF89AE1C892@wanadoo.fr> <53DBC493.5060700@dm.uba.ar> 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 1406976348 12496 80.91.229.3 (2 Aug 2014 10:45:48 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 2 Aug 2014 10:45:48 +0000 (UTC) Cc: Thomas Streicher , Categories To: "Eduardo J. Dubuc" Original-X-From: majordomo@mlist.mta.ca Sat Aug 02 12:45:42 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 1XDWoo-00087r-DJ for gsmc-categories@m.gmane.org; Sat, 02 Aug 2014 12:45:42 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49783) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XDWnv-0002Yg-KA; Sat, 02 Aug 2014 07:44:47 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XDWnu-0007Ou-22 for categories-list@mlist.mta.ca; Sat, 02 Aug 2014 07:44:46 -0300 In-Reply-To: <53DBC493.5060700@dm.uba.ar> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8270 Archived-At: Dear Eduardo, Of course I fully agree with what you said. Let me just add a few = remarks. =20 When Grothendieck defined fibrations it was for precise purposes, namely = to axiomatize the notion of inverse image, and to use this = axiomatization for descent (the titles of his talks are quite clear = about that) My view about fibrations is to try to get rid of sets, as much as = possible, in category theory. This is what I tried to explain in my = paper at the JSL. Thus if some fibrations arising from set theory come equipped with more = or less artificial cleavages, it does not impress me at all.=20 The same process of elimination of sets, makes very sensitive, for = fibrations but also for other domains, to the possibility of = internalization. Let me give an example of a totally different nature. In the paper where = I introduced bicategories, their definition is in section 1.1 and takes = 3 pages. But immediately after, it took me 10 pages to show in detail = that they could be internalized, in any category with pullbacks. Let me = point out that this paper was written in 1966 and published in 1967. I think of course that ZFC has been a huge progress in the history of = mathematics, but category theory has given us the possibility to explore = new and fascinating countries. This goes for many domains of = mathematics, and in particular.... for fibered categories. Best to all, Jean Le 1 ao=FBt 2014 =E0 18:47, Eduardo J. Dubuc a =E9crit : >=20 > Dear all, >=20 > I prefer fibrations over fibrations furnished with a cleavage (indexed = categories) many times for reasons purely pragmatical mixed with an = aesthetic philosophy. >=20 > Suppose you are dealing with fibrations where a canonical cleavage is = present, suppose even that these cleavages come first and that the = fibration is just a conceptual context around them. Even in this case, = faced to the need to produce a proof, if you succeed to find one without = utilizing the cleavages, you will have something much nicer than the = cleavage arguing. It will also give you a deeper understanding and a = truthful light on the situation. >=20 > Suppose you do not care about foundations, axiom of choice, or things = of that sort. You should still prefer fibrations. They are simpler, more = to the point, and CERTAINLY AHEAD IN THE PROGRESS OF MATHEMATICS. >=20 >=20 > NOTE: I wonder why so many people are so happy working with pull-backs = and pull-back preserving functors (*) without even thinking in = introducing a choice of pullbacks, and when it comes to fibrations, feel = the need to introduce and work with cleavages. >=20 > (*) for example even when dealing with the category of sets (or = categories whose objects have an underlying set), which are plenty of = choices of pull-backs, for example, inverse image of a subset, the = standard construction as a subset of the set of pairs, etc. We precisely = teach in category theory courses that you should not work with any = particular choice between the choices. >=20 > We all agree that it is neither necessary not good to choose a choice = between all possible choices. This is precisely the progress that = represents category theory thinking over set theory thinking. See for = example, in the dawn of category theory, the considerations of Mac Lane = concerning the fact that a quotient of a quotient of a group is not a = quotient (as it is still now taught in algebra courses, category theory = thinking has not arrived there yet). >=20 > You may say that the choice of a cleavage is at a different level than = all this, but, essentially, deep down, for me it is the same. There is = an old way of thinking (as Grothendieck said, SLN224, page 193) that = hesitates in face of fibrations and prefer to work with a chosen = cleavage. >=20 >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]