Dear Jean On 11/01/2011, at 6:31 PM, JeanBenabou wrote:
2- The situation is much worse in more general cases. Suppose E is a topos (this assumption is much too strong), and take C = Cat(E), the category of internal categories in E. On can define internal fibrations, and "fibrations" in the previous "abstract" sense. They do not coincide. It all boils down to the following remark: E and (E°, Set) are Toposes, the Yoneda functor E --> (E*,Set) preserves an reflects limits, but "nothing else" of the internal logic, which is needed to define internal fibrations.
I totally agree. An internal fibration between groups in a topos E is a group morphism whose underlying morphism in E is an epimorphism; for a representable fibration, it is a split epimorphism in E. Jack Duskin alerted me to this many years ago. Never-the-less, the representable notion has had some uses. Actually, Dominic Verity and I also used representably Giraud-Conduché morphisms in The comprehensive factorization and torsors, Theory and Applications of Categories 23(3) (2010) 42-75; whereas there is an internal version (more generally applicable in the way you explain) of these too (in a topos, for example). Have you written or published anything on these internal notions? Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]