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--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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]