Dear Jean, 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"). 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). Best regards, George -------------------------------------------------- From: "Jean B?nabou" <jean.benabou@wanadoo.fr> Sent: Monday, July 28, 2014 1:58 PM 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,

As i mentioned in my mail, it took me many years to develop foliated categories AND cartesian functors to their full extent. The very first approach starts circa 1984 when I had proved important results on cartesian functors between fibered categories and started wondering about possible generalizations. Thank you for sending me the paper of Cassidy, Herbert and Kelly which I do not know. I shall look at it carefully, but I doubt very much that it will have ANYTHING to do with foliated categories, let alone cartesian functors which are the essential content of my mail. I shall explain why after I have read the paper you are sending me.

Best regards to all, Jean

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