Re: Fibrations in a 2-category

[Ross Street <ross.street@mq.edu.au>]

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



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