[Paper Review] Fibred Categories \`a la Jean B\'enabou

Thomas Streicher|arXiv (Cornell University)|Jan 9, 2018

Homotopy and Cohomology in Algebraic Topology1 references7 citations

TL;DR

This paper provides a comprehensive exposition of Jean Bénabou's unpublished work on fibred categories, emphasizing their foundational role in category theory, topos theory, and categorical logic. It systematically explains how fibred categories generalize the notion of families of categories indexed over a base category using pullbacks, with the key contribution being a geometric and logical framework that unifies categorical structures across diverse mathematical domains.

ABSTRACT

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 The aim of these notes is to explain Bénabou’s approach to fi-bred categories which is mostly unpublished but intrinsic to most fields of category theory, in particular to topos theory and categorical logic.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 was

Motivation & Objective

• To clarify and disseminate Jean Bénabou’s unpublished yet influential approach to fibred categories, which underpins modern developments in category theory.
• To address the lack of accessible expositions of Bénabou’s geometric and logical treatment of fibred categories, especially in the context of base categories with pullbacks.
• To provide a coherent and systematic presentation of fibred categories based on Bénabou’s lectures and personal tutorials, as well as handwritten notes from his 1980 course at Louvain-la-Neuve.
• To integrate insights from J.-L. Moens’ 1982 thesis to extend the framework to more advanced categorical constructions.

Proposed method

• Adopting a geometric perspective, the paper constructs fibred categories as functors p: E → B satisfying the unique lifting property for cartesian morphisms over arbitrary base categories B with pullbacks.
• It formalizes the notion of a fibration using the universal property of cartesian lifts, ensuring that each morphism in the base category can be uniquely lifted to a morphism in the total category.
• The approach emphasizes the logical significance of fibred categories by interpreting them as indexed categories, enabling a natural treatment of quantifiers and type dependencies in categorical logic.
• It leverages pullbacks in the base category to define the structure of fibres and the way they relate across different base objects, ensuring coherence in the fibration.
• The paper draws on Bénabou’s personal insights and unpublished lectures to present a unified view of fibred categories that integrates both structural and logical aspects.
• It incorporates results from Moens’ thesis to extend the theory to include properties like full completeness and the existence of adjoints in fibred settings.

Experimental results

Research questions

• RQ1 How can fibred categories be systematically developed using only the structure of pullbacks in an arbitrary base category?
• RQ2 What is the precise role of cartesian morphisms in unifying geometric and logical structures in category theory?
• RQ3 How does Bénabou’s approach to fibred categories provide a foundation for categorical logic and topos theory?
• RQ4 In what ways do fibred categories generalize the notion of families of categories indexed over a base category?
• RQ5 How can the unpublished work of Jean Bénabou be made accessible and coherent for contemporary category theory research?

Key findings

• The paper establishes that fibred categories over a base category with pullbacks provide a natural framework for modeling dependent types and logical quantifiers in categorical logic.
• It demonstrates that the universal property of cartesian lifts ensures a canonical and unique way to transfer structure across fibres, enabling coherent indexing of categories.
• The fibration structure allows for a clean formulation of adjoint functors between fibres, particularly through the Beck-Chevalley condition and its variants.
• The integration of Moens’ results confirms that fibred categories support full completeness and the existence of adjoints under suitable conditions.
• The exposition reveals that Bénabou’s approach, though unpublished, is foundational and intrinsically linked to core developments in topos theory and categorical logic.