Re: Composition of Fibrations

["Jean Bénabou" <jean.benabou@wanadoo.fr>]

Dear Steve,

At least one ambiguity is solved. Chevalley gave as criterium (for opfibrations) that the arrow 
p~ : ΦE -> p/B  in your mail has a left adjoint with unit the identity.
When the 2-category is Cat this condition is satisfied iff  p is an opfibration which has an opcleavage. The choice of the adjoint defines the opcleavage.

Let us for the sake of precision call  Street criterium the existence of a left adjoint with unit an iso, and Street opfibrations (in Cat) the functors which satisfy this condition. 
They need not be opfibrations in the sense of Grothendieck which is almost unanimously adopted. It is unfortunate to have given them the name of (op)fibrations, not only because of the ambiguity as we have seen, but because the fibers are meaningless, in particular the fibers over two isomorphic objects of the base B need not be isomorphic. 

I'm almost sure that Neil Ghani, Richard Garner, Claudio Hermida and Thomas Streicher meant Grothendieck fibrations, and the genuine Chevalley condition in your answer, as I did.

Regards,

Jean

 

> Dear Jean,
> 
> Street's result is as follows. The arrow p: E -> B is a 0-fibration over B if and only if the arrow
>  p~ : ΦE -> p/B
> corresponding to the 2-cell
> 
> ΦE  --pd1--> B
> |                      ||
> d0                   ||
> |         pλ  =>   ||
> v                     ||
> E   --p------> B
> 
> has a left adjoint with unit an isomorphism.
> 
> Here ΦE = E/E and p/B are comma objects, d0 and d1 are projections, and λ is the canonical 2-cell in a comma square (in this case for ΦE). 0-fibration is opfibration.
> 
> Regards,
> 




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