FIBRED CATEGORIES a ̀ la Jean Bénabou

purely geometric reasons. The “logical ” aspect of fibred categories and, in particular, their relevance for category theory over an arbitrary base cate-gory with pullbacks has been investigated and worked out in detail by Jean Bénabou. The aim of these notes is to explain Bénabou’s approach to fi-bred categories which is mostly unpublished but intrinsic to most fields of category theory, in particular to topos theory and categorical logic. There is no claim for originality by the author of these notes. On the contrary I want to express my gratitude to Jean Bénabou for his lectures and many personal tutorials where he explained to me various aspects of his work on fibred categories. I also want to thank J.-R. Roisin for making me available his handwritten notes [Ben2] of Des Catégories Fibrées, a course by Jean Bénabou given at the University of Louvain-la-Neuve back in 1980. The current notes are based essentially on [Ben2] and a few other insights of J. Bénabou that I learnt from him personally. The last four sections are based on results of J.-L. Moens ’ Thése [Moe] from 1982 which itself wa