Re: Composition of Fibrations

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

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,

Steve.

> On 21 Jul 2014, at 19:02, Jean Bénabou <jean.benabou@wanadoo.fr> wrote:
> 
> Dear Steve,
> 
> Thank you for your prompt  answer. 
> 
> Let me first clarify a possible ambiguity. 
> The Street fibrations I was referring to are defined in his paper: 
> Fibrations in bicategories.  Cahiers Top. Geom. Diff. 21 (1980)
> When the bicategory is Cat, they do not coincide with the usual fibrations. In particular every equivalence is a Street fibration.
> 
> There might be another ambiguity about what you call the Chevalley criterium. Could you please tell me with precision what it is (I assume p: B -> A is a map in a 2-category C with comma objects and 2-pullbacks)
> 
> I shall come back to mathematical questions as soon as these two ambiguities are solved.
> 
> Best wishes,
> 
> Jean
> 

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