Re: A brief survey of cartesian functors

["George Janelidze" <janelg@telkomsa.net>]

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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