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On the Bénabou-Roubaud theorem

Bruno Kahn

TL;DR

The paper provides a detailed proof of the Bénabou–Roubaud theorem in its original setting and shows that its hypotheses can be weakened: the base category need not have fibre products and the Beck–Chevalley condition can be weakened to a requirement that base-change morphisms be epi. It develops adjoint-chase techniques to analyze base-change morphisms $\chi$ and relates exchange to the Chevalley condition (C) and its dual, establishing equivalences and essential-surjectivity results for the Eilenberg–Moore comparison functor $K^a$ under strengthened unit-descent hypotheses. By introducing pre-descent data and the unit condition via a diagonal $\Delta$, it proves an equivalence between the algebraic and descent-data perspectives under a technical hypothesis $h_1$, and it provides a supplement showing associativity of unital $T^a$-algebras under fully faithful pullback functors accompanied by Karoubi completeness. The appendix gives a concrete case where the exchange condition holds (notably reproducing Mackey-type formulas in the group context) and discusses how these results extend to presheaf categories and representations in categorical settings, with implications for base-change and descent theory.

Abstract

We give a detailed proof of the Bénabou-Roubaud theorem. As a byproduct it yields a weakening of its hypotheses: the base category does not need fibre products and the Beck-Chevalley condition, in the form of a natural transformation, can be weakened by only requiring the latter to be epi.

On the Bénabou-Roubaud theorem

TL;DR

The paper provides a detailed proof of the Bénabou–Roubaud theorem in its original setting and shows that its hypotheses can be weakened: the base category need not have fibre products and the Beck–Chevalley condition can be weakened to a requirement that base-change morphisms be epi. It develops adjoint-chase techniques to analyze base-change morphisms and relates exchange to the Chevalley condition (C) and its dual, establishing equivalences and essential-surjectivity results for the Eilenberg–Moore comparison functor under strengthened unit-descent hypotheses. By introducing pre-descent data and the unit condition via a diagonal , it proves an equivalence between the algebraic and descent-data perspectives under a technical hypothesis , and it provides a supplement showing associativity of unital -algebras under fully faithful pullback functors accompanied by Karoubi completeness. The appendix gives a concrete case where the exchange condition holds (notably reproducing Mackey-type formulas in the group context) and discusses how these results extend to presheaf categories and representations in categorical settings, with implications for base-change and descent theory.

Abstract

We give a detailed proof of the Bénabou-Roubaud theorem. As a byproduct it yields a weakening of its hypotheses: the base category does not need fibre products and the Beck-Chevalley condition, in the form of a natural transformation, can be weakened by only requiring the latter to be epi.
Paper Structure (6 sections, 15 theorems, 33 equations)

This paper contains 6 sections, 15 theorems, 33 equations.

Table of Contents

  1. Adjoint chases
  2. Exchange condition and weak exchange condition
  3. Pre-descent data
  4. The unit condition
  5. A supplement
  6. Appendix: a case where the exchange condition is verified

Key Result

Lemma 1.2

For any $\varphi\in \mathbf{M}(A_1)(T^a M,N)$, one has

Theorems & Definitions (34)

  • Remark 1.1
  • Lemma 1.2: key lemma
  • proof
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • Proposition 1.6
  • proof
  • Lemma 1.7
  • ...and 24 more