Re: A brief survey of cartesian functors

[Paul Levy <P.B.Levy@cs.bham.ac.uk>]

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  iff 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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