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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]