From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8255 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: Re: A brief survey of cartesian functors Date: Tue, 29 Jul 2014 11:16:20 +0200 Message-ID: References: <1B862C69106C4B6A83703605D3E6A693@ACERi3> 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 1406682435 31351 80.91.229.3 (30 Jul 2014 01:07:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 30 Jul 2014 01:07:15 +0000 (UTC) Cc: "Categories" To: "George Janelidze" Original-X-From: majordomo@mlist.mta.ca Wed Jul 30 03:07:10 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 1XCIMG-00085z-Tr for gsmc-categories@m.gmane.org; Wed, 30 Jul 2014 03:07:09 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49097) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XCILy-0005VC-ML; Tue, 29 Jul 2014 22:06:50 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XCILz-00082U-5J for categories-list@mlist.mta.ca; Tue, 29 Jul 2014 22:06:51 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8255 Archived-At: Dear George, I appreciate very much your pioneer work on Galois theories and the = developments you and others have given to that work. I also believe in the role of analogies in mathematics, and I think = category theory is the ideal place where one can give the DEEP analogies = a mathematical content. However, in this case, the analogy seems to me totally superficial, = namely: two classes A and B of maps in a category X, and the possibility = to factor every map f of X as ab, with a in A and b in B.=20 This won't go very far since you need some axioms on the pair (A,B) to = start proving anything except trivialities. And, I tried to explain in = my previous mail, the properties of pairs (E,M) and (V,K) are so = radically different that a common denominator would be reduced to almost = nothing. Even more important to me, cartesian functors are a very good notion of = morphism between pairs (V,K) and (v',K') which you can prove non = trivial results, the theorem in my mail is only an example of such = results. As far as I know there is no notion of morphism between pairs = (E,M) and (E',M'). Let me point out some features of cartesian functors F: X --X' , viewed = abstractly as morphisms (V,K) --> (V',K') where V =3D V(P), K =3D = K(P), V' =3D V(P') and K' =3D K(P'). 1) F preserves vertical end cartesian maps. This is harmless, but F also = REFLECTS vertical maps. 2) We assume that every map of X can be factored as kv, but we make no = such assumption on X' 3) The very nature of the results: For any important properties, F = satisfies globally the property iff it satisfies it fiberwise. If any reasonable notion of morphism of pairs (E,M) was defined someday = would reflection of maps in M be considered? Would one accept that = (E',M') should not be a factorization system even in a very weak sense? And if non trivial results could be obtained about such notion would = some kind of fibers play a role? Sorry George, much as I like unifying notions and theories, I cannot see = any real, non trivial, relation between factorization systems and (pre = folations + cartesian functors) I insist on the second term of the previous symbolic addition. There would be a lot more to say about indexed versus fibered, but you = already know my opinion about that. Moreover indexed is totally = irrelevant here becausethere is no reindexing for pre foliations Best regards, Jean Le 29 juil. 2014 =E0 09:02, George Janelidze a =E9crit : > Dear Jean, >=20 > Talking about the comparison, I had in mind mainly the following: the = vertical-cartesian factorization for a fibration is closely related to = the reflective factorization system for a semi-left-exact reflection = (one might vaguely say "they are the same up to an isomorphism under the = assumptions used in both of them"). >=20 > Concerning the older discussion on fibrations versus indexed = categories: Please believe me that I fully agree with every instance of = "fibrations are better" you mention. Nevertheless I also agree with = "indexed categories are better", in a different sense. The reason I am = saying this now is that I would like to mention semi-left-exact = reflections of Cassidy--Hebert--Kelly and their generalizations as a = THIRD APPROACH (I used them independently calling them "admissible" in = Galois theory, first exactly in 1984). >=20 > Best regards, > George [For admin and other information see: http://www.mta.ca/~cat-dist/ ]