Re: Composition of Fibrations

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

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


Le 21 .. 2014 à 14:30, Steve Vickers a écrit :

> Dear Jean,
> 
> Thank you for your detailed comments.
> 
> Something I should say straight away is that the duality argument I had in mind, dualizing 2-cells, might be OK to deal with left adjoints to reindexing but was completely wrong for right adjoints. Already, Richard Garner and Claudio Ermida (thanks to both of them) have shown me that it doesn't do the job.
> 
> I also want to stress that at no point did I intend to set up my own definition of fibration. I was following Street's "Fibrations and Yoneda's lemma in a 2-category", which defines fibrations as those 1-cells that carry pseudoalgebra structure for a certain 2-monad, and then proves (Proposition 9) that this is equivalent to what Street refers to as the Chevalley condition. If "Vickers' definition" is not equivalent to that then I have made a mistake somewhere. 
> 
> Have you found a discrepancy between the "Vickers definition" and Street? At one point you write "Of course, I don't refer here to Street's notion which describes a totally different kind of fibration, stable by equivalences."
> 
> I agree that the concept I have been using includes cleavage (and, for a bifibration, cocleavage). I cannot assume AC in what I do, and I rather imagined that structure something like the Chevalley criterion was needed in order to deal with its absence. However, I admit I am not so familiar with the fully general notion of fibration. For me the Chevalley condition seemed enough to do what I needed in the 2-category Loc of locales and my remarks were based on that experience.
> 
> Best wishes,
> 
> Steve.
> 
> 
> 

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