30 Jul 03:05 2014

### Re: A brief survey of cartesian functors

Jean Bénabou <jean.benabou <at> wanadoo.fr>

2014-07-30 01:05:30 GMT

2014-07-30 01:05:30 GMT

Dear George, Thank you for your mail. I see that all my mathematical arguments have not convinced you, and that trying to add more would be useless. I respect your opinion although I totally disagree with it. Best regards, Jean Le 29 juil. 2014 à 21:58, George Janelidze a écrit : > 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--(Continue reading)~~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 >