From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8257 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: A brief survey of cartesian functors Date: Tue, 29 Jul 2014 21:58:05 +0200 Message-ID: References: <1B862C69106C4B6A83703605D3E6A693@ACERi3> <54F4E17E-FAD3-43D8-89F2-5B9CF1C098D8@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 1406751734 13185 80.91.229.3 (30 Jul 2014 20:22:14 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 30 Jul 2014 20:22:14 +0000 (UTC) Cc: "Categories" To: =?iso-8859-1?Q?Jean_B=E9nabou?= Original-X-From: majordomo@mlist.mta.ca Wed Jul 30 22:22:08 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 1XCaNv-0004CA-Fc for gsmc-categories@m.gmane.org; Wed, 30 Jul 2014 22:22:03 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49224) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XCaNQ-0000CP-0m; Wed, 30 Jul 2014 17:21:32 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XCaNP-0002iO-Vh for categories-list@mlist.mta.ca; Wed, 30 Jul 2014 17:21:31 -0300 In-Reply-To: <54F4E17E-FAD3-43D8-89F2-5B9CF1C098D8@wanadoo.fr> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8257 Archived-At: Dear Jean, Thank you for your kind words at the beginning of your message, and I apologize if what I said about "factorization" and "cartesian" was unclear. I did not mean to say that there is any connection between factorization systems and (pre foliations + cartesian FUNCTORS). What I was trying to say, was only that the following two constructions are essentially the same (up to an isomorphism): (a) For a fibration C-->X every morphism f in C factors as f = me, where m is a cartesian ARROW and e is a vertical arrow (with respect to the given fibration). (b) For a semi-left-exact reflection C-->X (in the sense of Cassidy--Hebert--Kelly) every morphism f in C factors as f = me, where m is in M, e is in E, E is the class of all morphisms inverted by C-->X, and M is its orthogonal class (M can also be defined as the class of trivial covering morphisms in the sense of Galois theory). I know this might sound trivial to you, but I think it is a fundamental connection, which should be widely known. And I believe that instead of "indexed categories versus fibrations" one should sometimes also consider "indexed categories versus fibrations versus semi-left-exact reflections" (this is why I mentioned a "third approach"). Let me also add now: according to Cassidy--Hebert--Kelly, the factorization mentioned in (b), where E is as in (b), and M is merely its orthogonal class, also exists under certain assumptions much weaker than semi-left-exactness. But again, I never thought that what you do with pre foliations and cartesian functors is a similar kind of factorization and/or that it is contained in the Cassidy--Hebert--Kelly paper! And I hope you have never felt from me any disrespect of your opinions and/or of your beautiful ideas and results. Best regards, George -------------------------------------------------- From: "Jean B?nabou" Sent: Tuesday, July 29, 2014 11:16 AM To: "George Janelidze" Cc: "Ross Street" ; "Steve Vickers" ; "Lack Steve" ; "Peter Johnstone" ; "Eduardo Dubuc" ; "Thomas Streicher" ; "Robert Par?" ; "Marta Bunge" ; "William Lawvere" ; "Michael Wright" ; "Categories" Subject: Re: A brief survey of cartesian functors > 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. > > 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 = V(P), K = K(P), V' > = V(P') and K' = 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 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]