Renewed study of these ideas, in light of the later exposition by Thomas Streicher, should lead to further applications (for example to the solution of problems posed in my 1972 Perugia Notes.)
Thanks for mentioning this text. But since I think ownership of ideas is an important issue I want to point out that the aim of this text was to exhibit part of the ideas and results of Jean B'enabou's almost single handed approach to Fibered Categories as a foundation of Category Theory over most general base categories. I also tried to explain some work by J.-L. Moens (his 1982 Thesis) where I think I have added a bit of additional material. As far as B'enabou's work is concerned my sources were Roisins notes from B'enabou's 1980 Louvain-la-Neuve lectures. My exposition evolved over the years and I have integrated additional material and made corrections as I learnt from Jean for which I am very grateful to him. At one place I have referred to a fibred version of the Special Adjoint Functor Theorem which one can find in Par'e and Schumacher's text or alternatively in J.Celeyrette's These d'Etat from 1974 under supervision of B'enabou. The references in my text are not exhaustive at all. It's certainly a mistake not to have formally referred to work by Grothendieck and Giraud. But I was not using too much these original sources. Moreover, I have used results from a paper by Mamuka Jibladze. I have not given the precise reference but made clear in the title of the appendix that it is Mamuka's result. The aim of these notes was not to document precisely who did what. However, on the first page I clearly stated whose work influenced me! I am also aware that in Bill's "Perugia Notes" one can find the idea that fibered or indexed categories are suitable for doing category theory over a base topos. But as I said it wasn't my intention to document the history of ideas but rather to write up the view of things as I learnt it from Jean's work and private communications. Thomas Streicher [For admin and other information see: http://www.mta.ca/~cat-dist/ ]