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From: Paul Levy
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Subject: Re: A brief survey of cartesian functors
Date: Fri, 1 Aug 2014 11:35:58 +0100
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On 28 Jul 2014, at 10:54, Jean B=E9nabou wrote:
> 2) CARTESIAN FUNCTORS
> Let P: X --> S, P': X' --> S and F: X --> X' be functors such =20
> that P =3D P'F. For every object s of S ,I denote by F_s : X_s --> =20=
> X'_s the functor induced by F on the fibers.
> I have a general definition of F being cartesian, without any =20
> assumption on P and P' and without any reference to cartesian maps, =20=
> 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 =20
setting?
Paul
> 2.3. THEOREM. If P is a pre foliation, P' arbitrary, and F is =20
> 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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