Thanks for that "ungated" link to Bénabou's 1985 JSL paper, Wes.
As it turned out, I'd downloaded all 28 pages back in 2011, I can't remember why. Moreover it arrived 26 years earlier in my mailbox in 1985, but who reads every article, especially if you're more on the universal algebra side than the category theory side in 1985.
Now that it's 2024, what I'm wondering is, is there 28 pages of useful information there?
I appreciate the idea of a fibration as a functor C → B where B is a "base category" looking enough like Set (aka Ens). In recent years I've been writing about the Yoneda embedding as a way of viewing Chu spaces over Set, namely as Set-valued functors representing structured objects. This sort of thing is closer to toposes than abelian categories, both of which I like a lot given that the Big Bang seems to have evolved from a topos to (at least in our neighborhood) an abelian category.
What I'm having difficulty with is why Jean took 28 pages to make his point. It seemed better suited to the Journal of Philosophical Logic than to JSL.
I'm a great fan of Einstein's "make everything as simple as possible but no simpler". However that implies that there's a boundary, and in the case of fibrations and definability, I'm not at all clear as to where that boundary should be drawn.
Vaughan Pratt