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

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