Re: A brief survey of cartesian functors

["Jean Bénabou" <jean.benabou@wanadoo.fr>]

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



Le 29 juil. 2014 à 09:02, George Janelidze a écrit :

> 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



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