Fibrations in a 2-category

[JeanBenabou <jean.benabou@wanadoo.fr>]

I have seen very often the following "abstract" definition of a  
fibration in a 2-category C :
A map (i.e. a 1-cell) p: X --> S is a fibration iff for each object Y  
of C the functor C(Y,p):  C(Y,X) --> C(Y,S) is a fibration (in the  
usual sense) which depends "2-functorially" on Y.

Such an "obvious" definition is much too naive and does not give the  
correct notion in most examples.

1- Even if C= Cat, the 2-category of (small) categories, a fibration  
in the abstract sense is a Grothendieck fibration which admits a  
cleavage. Thus if we don't assume AC, which we don't need to define  
fibrations, it does not coincide with the usual one.

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.

Best to all,

Jean
   

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