From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8262 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: Wed, 30 Jul 2014 19:56:54 +0200 Message-ID: <89048344-26F3-448F-8B41-9FF89AE1C892@wanadoo.fr> References: <20140730150643.GC19613@mathematik.tu-darmstadt.de> 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 1406857140 22594 80.91.229.3 (1 Aug 2014 01:39:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 1 Aug 2014 01:39:00 +0000 (UTC) Cc: Categories To: Thomas Streicher Original-X-From: majordomo@mlist.mta.ca Fri Aug 01 03:38:51 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 1XD1o1-0005EL-Vo for gsmc-categories@m.gmane.org; Fri, 01 Aug 2014 03:38:50 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49449) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XD1nj-0004u4-I7; Thu, 31 Jul 2014 22:38:31 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XD1nk-0007e5-AX for categories-list@mlist.mta.ca; Thu, 31 Jul 2014 22:38:32 -0300 In-Reply-To: <20140730150643.GC19613@mathematik.tu-darmstadt.de> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8262 Archived-At: Dear Thomas, I am a bit surprised that you, of all people, should defend cleavages, = i.e. indexed categories. As far as I remember, there not many of them in = the notes you wrote on Fibered categories a la Benabou. I remind you that these notes were written not only after lectures I = gave, but after long conversations we had, many times, in my flat, and = also a few days you spent in my house in the south of France, where, for = at least 10 hours everyday I explained in detail to you my work on = fibered categories and corrected many mistakes you made in first drafts = of that paper. Nevertheless, for the sake of the people on the category list,to whom = this message is also addressed, I shall answer your questions an = remarks. Let p: X --> S be a surjective group homomorphism. It is a fibration, = and a cleavage is a section of p (in Set). This was explicitly noted by = Grothendieck more than 50 years ago!=20 Does such a p come equipped with a cleavage? Take for p the morphim : = R --> R/Z of the reals on the circle. Suppose I were to teach periodic = functions, i.e functions with domain R which factor through R/Z. = Wouldn't it be ridiculous to use a section of p ? Which one by the way? Take the theorem : The composite of two fibrations is a fibration. Does = it need cleavages, i.e. AC for classes, to be proved? Of course, if you = are cleavage fan, as you seem to be, you can add that, given cleavages = of p and q one gets an associated cleavage of pq. Let's look at an important example,namely categories S with pull backs, = not with choice of pull backs mind you. This is a first order notion, = saying that : for every cospan of S there exists a universal span making = the obvious square commutative. Then, without AC, you can prove that the functor Codom: S^2 --> S is a = fibration. And again, if you are cleavage happy, add that a cleavage of = this fibration, if it exists, (I'm not assuming AC) is a choice of = pullbacks in S. I could multiply the examples. But let's look at an important question. = Suppose you prove an intrinsic result about fibrations, using cleavages, = in principle you'd have to see what happens when you change cleavages. = And don't wave your hands and tell me that, for formal 2-categorical = reasons, the result is obvious. I'll believe you only when you write a = precise metatheorem which covers ALL the cases. You are convinced, and I am convinced, and everybody is convinced that = such a metatheorem is not necessary. But that is NOT A PROOF ! And why = are you convinced? Because, even if you say the contrary, deep in your = mind you KNOW that intrinsic properties of fibrations AND cartesian = functors should not refer to cleavages. Let me insist on the fact that = the mythic metatheorem should also cover cartesian functors F: X --> X' = where you change the cleavages of both X and X' Of course the theorem I mentioned in my mail on pre foliations, applies = to fibrations and gives new results in that case. But this theorem is = true for F: X --> X' where X is a prefoliation hence, even with AC, = has no cleavage, and X' is an arbitrary category over S, i.e. has even = less cleavages than X. In order not to make this mail too long I have not, but I should have, = mentioned internalization where cleavages are even more problematic. Best to all, Jean Le 30 juil. 2014 =E0 17:06, Thomas Streicher a =E9crit : > Dear Jean, >=20 > of course, you are right when emphasizing that one need choice for > classes to endow an "anonymous" fibration with a cleavage. > But that applies also to catgeories with say binary products. One > needs choice for classes in order to choose a product cone for every > pair of objects. > In many instances, however, categories come together with a choice of > products and fibrations come together with a choice of a cleavage.=20 >=20 > For example Set comes with a choice of a cleavage. Fibrations arising > from internal categories are even split. Many constructions on = fibrations=20 > allow one to choose a cleavage given cleavages for the arguments. > Do you know of any construction on fibrations which is not "cleavage > preserving" in this sense? >=20 > Of course, one should not require cartesian functors to preserve > cleavages just as one should not require functors to preserve chosen > products. >=20 > Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]