From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8277 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Present and future Date: Sun, 3 Aug 2014 17:44:03 +0200 Message-ID: References: <2A04D350-E841-4273-8175-95190111016B@wanadoo.fr> Reply-To: "George Janelidze" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1407156548 31756 80.91.229.3 (4 Aug 2014 12:49:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 4 Aug 2014 12:49:08 +0000 (UTC) To: "Categories" Original-X-From: majordomo@mlist.mta.ca Mon Aug 04 14:49:01 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 1XEHhE-0005Yy-MS for gsmc-categories@m.gmane.org; Mon, 04 Aug 2014 14:49:00 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50068) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XEHgs-0007WW-IZ; Mon, 04 Aug 2014 09:48:38 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XEHgr-0002pU-DC for categories-list@mlist.mta.ca; Mon, 04 Aug 2014 09:48:37 -0300 In-Reply-To: <2A04D350-E841-4273-8175-95190111016B@wanadoo.fr> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8277 Archived-At: Dear Jean, I don't want us to live on different planets - so, I am making one more attempt: My feeling is that you interpret everything I say as "some kinds of mathematical objects are better than fibrations" ("some kinds" could be indexed categories, or pseudo-fibrations, or, say, semi-left exact reflections). And then you give convincing examples where the language of fibrations works better, and then you say that you could not convince me. But: I NEVER said that any of those concepts is better! All I was trying to say (more than once) is that all of them, including fibrations, are very important. Moreover, the relationship between them - which is not exactly an equivalence - is a very serious mathematical result/discovery/idea, which, as well as as some other ideas of category theory, helps us to see better the whole planet of mathematics (on which all recipients of this message live, I suppose). By the way, a very 'small part' of the relationship between fibrations and indexed categories, namely the equivalence between discrete fibrations over a category C (with small fibres) and functors C^op-->Sets, is already a fundamental result, is not it? Well, working with discrete fibrations eliminates sets to a larger extend: e.g. we don't need to think of small fibres, and we can internalize them (I mean, define discrete fibrations over an internal category). But does it mean that we should forget about Set-valued functors? I know everything I said is trivial for you, but, forgive me, you forced me. Best regards to all, George -------------------------------------------------- From: "Jean B?nabou" Sent: Saturday, August 02, 2014 6:00 PM To: "Thomas Streicher" ; "Eduardo Dubuc" ; "Categories" Subject: categories: Present and future > Dear Thomas, > > As I told you in my previous private mail you are entitled to have your > own view, and to make it public. You don't have to submit me anything. I > shall of course respect your opinion, even if I disagree with it. (by the > way I told exactly the same thing to George Janelidze but, not only I > could not convince him, but I had the impression we were living on > different planets!). > Of course, if I do disagree, I shall tell you why I do, and try to > convince you by purely mathematical arguments, not by the fact that I > consider myself as some kind of owner of fibered categories, in spite of > the important developments of this theory which I introduced. > And I promise to study carefully your own arguments,c and to change my > views about some questions if you convince me, mathematically. > > This is by no means an an answer to your mail. I am preparing a more > ambitious mail, where I shall expose my views, not only about fibrations > but on other important issues, some of which have not, or very little, > been touched by the numerous mails about fibrations exchanged during the > last weeks. > Because of the comprehensive scope of this future mail, I beg you to be > patient, i shall need some time. > This future mail shall, in a sense, be addressed to me. I'm getting old, > and I need to think a little about what I have done, and what I should > have done. (Not only in mathematics of course, but the other domains are > between me and me). > > Best to all, > Jean > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]