Fibered Categories a la Jean Benabou

TL;DR

This paper provides an overview of the theory of Fibred Categories based on Jean Benabou's work, highlighting key notions and their intuitive explanations, with historical context and personal insights.

Contribution

It compiles and explains the core ideas of fibered categories as developed by Benabou, including insights from related works, in a mostly unpublished and informal manner.

Findings

Clarifies fundamental concepts of fibered categories.

Connects fibered categories to geometric morphisms.

Provides a personal interpretation of Benabou's work.

Abstract

These are notes about the theory of Fibred Categories as I have learned it from Jean Benabou. I also have used results from the Thesis of Jean-Luc Moens from 1982 in those sections where I discuss the fibered view of geometric morphisms. Thus, almost all of the contents is not due to me but most of it cannot be found in the literature since Benabou has given many talks on it but most of his work on fibered categories is unpublished. But I am solely responsible for the mistakes and for misrepresentations of his views. And certainly these notes do not cover all the work he has done on fibered categories. I just try to explain the most important notions he has come up with in a way trying to be as close as possible to his intentions and intuitions. I started these notes in 1999 when I gave a course on some of the material at a workshop in Munich. They have developed quite a lot over the years and I have tried to include most of the things I want to remember.