R A brief survey of cartesian functors

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

Dear Paul,

I have received many mails concerning the posting you mentioned and I try to answer all of them with precision. This takes a lot of time. I shall answer yours, in detail, about the general definition of cartesian functors in terms of disdtributors very soon. But the precise answer shall be a bit long, so please be patient. 
All I can say now is that, if P: X --> S is a prefoliation the general definition reduces to the simple one I have given (the proof is not trivial).
Moreover some of the results of the theorem require the assumption that P is a prefoliation. Hardly surprising, if you want to prove stronger results you need stronger assumptions. The ones I make are, in a sense, minimal. You'll have I think noticed that for 7) I need P to be a foliation.

Thanks for your interest. Best regards,

Jean


Le 1 août 2014 à 12:35, Paul Levy a écrit :

> 
> On 28 Jul 2014, at 10:54, Jean Bénabou wrote:
> 
>> 2) CARTESIAN FUNCTORS
>> Let  P: X --> S,  P': X' --> S  and  F: X --> X' be functors such that  P = P'F.  For every object s of S ,I denote by  F_s : X_s --> X'_s  the functor induced by  F  on the fibers.
>> I have a general definition of F being cartesian, without any assumption on P and P' and without any reference to cartesian maps, but it uses distributors in an essential manner.
> 
> Please tell us your general definition using distributors.
> 
> Do any of the results in your Theorem 2.3 hold in this more general setting?
> 
> Paul
> 
>> 2.3.  THEOREM. If P is a pre foliation, P' arbitrary, and F is cartesian, then:
>> (1)  F is faithful    every  F_s  is.
>> (2)  F is full iff every F_s   is.
>> (3)  F is essentially surjective iff every  F_s  is.
>> (4)  F is final iff every  F_s  is.
>> (5)  F is flat iff every  F_s  is.
>> (6)  F has a left adjoint iff every  F_s  has.
>> If moreover P is a foliation, then
>> (7)  F is conservative iff every  F_s  is.
> 
> --
> Paul Blain Levy
> School of Computer Science, University of Birmingham
> +44 121 414 4792
> http://www.cs.bham.ac.uk/~pbl

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