These are notes about the theory of Fibred Categories as I have learned it
from Jean Benabou. I also have used results from the Thesis of Jean-Luc Moens
from 1982 in those sections where I discuss the fibered view of geometric
morphisms. Thus, almost all of the contents is not due to me but most of it
cannot be found in the literature since Benabou has given many talks on it but
most of his work on fibered categories is unpublished. But I am solely
responsible for the mistakes and for misrepresentations of his views. And
certainly these notes do not cover all the work he has done on fibered
categories. I just try to explain the most important notions he has come up
with in a way trying to be as close as possible to his intentions and
intuitions. I started these notes in 1999 when I gave a course on some of the
material at a workshop in Munich. They have developed quite a lot over the
years and I have tried to include most of the things I want to remember.Comment: Have added an appendix describing a fibrational account of Lawvere's
notion of (stably) precohesive geometric morphism