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" <jean.benabou@wanadoo.fr> Sent: Tuesday, July 29, 2014 11:16 AM To: "George Janelidze" <janelg@telkomsa.net> Cc: "Ross Street" <street@ics.mq.edu.au>; "Steve Vickers" <s.j.vickers@cs.bham.ac.uk>; "Lack Steve" <steve.lack@mq.edu.au>; "Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>; "Eduardo Dubuc" <edubuc@dm.uba.ar>; "Thomas Streicher" <streicher@mathematik.tu-darmstadt.de>; "Robert Par?" <pare@mathstat.dal.ca>; "Marta Bunge" <martabunge@hotmail.com>; "William Lawvere" <wlawvere@hotmail.com>; "Michael Wright" <mpbw1879@yahoo.co.uk>; "Categories" <categories@mta.ca> 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

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