[Ross: Any comments?] Dear Jean, I wasn't properly aware of those issues around choosing iso or equality, so it's lucky I got into this discussion. My intuition of what you are saying is that with iso, roughly speaking, the reindexing is only pseudofunctorial. For example, if you pull back along the diagonal B -> ΦB, to get the reindexing along identities, then you get an endofunctor of each fibre that is isomorphic to the identity. Am I on the right lines? Is this all written down somewhere? After your messages I noticed that Street has a remark after his proposition, whose significance I overlooked: "Compare the above proposition with Gray [2] p.56; so we have related the definition of 0-fibration here with the definition of opfibration in [2] when K = Cat. Notice that the unit of the adjunction l -| p~ for Gray is not just an isomorphism but an identity. It is worth pointing out the reason for this since we will need the observation in the next paper. A 0-fibration will be called normal when there is a normalized pseudo L-algebra structure on it. In Cat every 0-fibration is normal, but in other categories this need not be the case. In the proof of the Chevalley criterion, if ζ is an identity then so is η. So, for a normal 0-fibration, p~ : ΦE -> p/B has a left adjoint with unit an identity." Notes: 1. Gray [2] = "Fibred and cofibred categories", La Jolla. 2. I don't know which paper Street means by "the next paper". 3. In a pseudo L-algebra E, with structure morphism c: LE -> E, ζ denotes the isomorphism from Id_E to unit composed with c. E is normalized if ζ is equality. 4. η is the unit of the adjunction. But that seems to claim that in Cat it doesn't matter whether you use iso or equality in the Chevalley condition. Does that accord with your understanding? Regards, Steve.
On 22 Jul 2014, at 05:24, Jean Bénabou <jean.benabou@wanadoo.fr> wrote:
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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