Absolute descent and the Bénabou--Roubaud theorem

Abstract

We continue the programme developed in Pseudo-Kan extensions and descent theory by studying descent for augmented pseudocosimplicial objects. Let ℋ be a $2$-category, and let suc=1+(−) be the successor monad on the augmented simplex category. Precomposition with suc induces the décalage pseudomonad 𝕊 on the $2$-category of augmented pseudocosimplicial objects with values in ℋ. We prove that every successor contraction makes the corresponding augmented pseudocosimplicial object effective for descent. This structure is preserved by postcomposition with every pseudofunctor. Consequently, the resulting descent object is an absolute bilimit. A first application concerns split epimorphisms. We recover the Janelidze--Tholen characterization of absolute effective descent morphisms given in Facets of Descent II. Moreover, we prove that one direction holds for every pseudofunctor A: C^op→ℋ, where ℋ is an arbitrary $2$-category. Consequently, every 𝕊-pseudoalgebra is of absolute descent. In particular, this holds for every free 𝕊-pseudoalgebra, and hence for every one-sided décalage. This makes the shift-effectiveness hypothesis in the Bénabou--Roubaud theorem of Pseudo-Kan extensions and descent theory automatic. Thus, under the adjointness and Beck--Chevalley hypotheses, the descent and Eilenberg--Moore factorizations are equivalent whenever they exist. We thus obtain a stronger version of the Bénabou--Roubaud theorem for arbitrary augmented pseudocosimplicial objects, which we call the formal Bénabou--Roubaud theorem. Applied to indexed Čech objects associated with pseudofunctors A: C^op→ℋ, this theorem extends the classical Bénabou--Roubaud theorem to an arbitrary $2$-category ℋ.

The multivariate Berry-Esseen theorem with an absolute constant
Zhi Ling
arXiv1 Nov 2026PreprintarXiv licenceFull textProbability StatisticsarXiv
Infinite free pro-p products as absolute Galois groups
Tamar Bar-On
arXiv1 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Iterative Methods for Solving New Generalized Absolute Value Equations
Shubham Kumar, Joydip Dhar
arXiv29 Oct 2026PreprintarXiv licenceFull textOptimisation & controlarXiv
Bad(r;s) is Hyperplane Absolute Winning
Chengyang Wu
arXiv18 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Dynamical systemsarXiv
Closed-form evaluation of the cumulative distribution function and absolute moments of the variance-gamma distribution
Robert E. Gaunt, Heather L. Sutcliffe
arXiv12 Oct 2026PreprintarXiv licenceFull textProbabilityarXiv

Cite

BibTeX RIS

BibTeX

@misc{nunes2026absolute,
  title = {{Absolute descent and the Bénabou--Roubaud theorem}},
  author = {Fernando Lucatelli Nunes},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01654},
  eprint = {2611.01654},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01654},
}

RIS

TY  - UNPB
TI  - Absolute descent and the Bénabou--Roubaud theorem
AU  - Fernando Lucatelli Nunes
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01654
UR  - https://arxiv.org/abs/2611.01654
AB  - We continue the programme developed in Pseudo-Kan extensions and descent theory by studying descent for augmented pseudocosimplicial objects. Let ℋ be a $2$-category, and let suc=1+(−) be the successor monad on the augmented simplex category. Precomposition with suc induces the décalage pseudomonad 𝕊 on the $2$-category of augmented pseudocosimplicial objects with values in ℋ. We prove that every successor contraction makes the corresponding augmented pseudocosimplicial object effective for descent. This structure is preserved by postcomposition with every pseudofunctor. Consequently, the resulting descent object is an absolute bilimit. A first application concerns split epimorphisms. We recover the Janelidze--Tholen characterization of absolute effective descent morphisms given in Facets of Descent II. Moreover, we prove that one direction holds for every pseudofunctor A: C^op→ℋ, where ℋ is an arbitrary $2$-category. Consequently, every 𝕊-pseudoalgebra is of absolute descent. In particular, this holds for every free 𝕊-pseudoalgebra, and hence for every one-sided décalage. This makes the shift-effectiveness hypothesis in the Bénabou--Roubaud theorem of Pseudo-Kan extensions and descent theory automatic. Thus, under the adjointness and Beck--Chevalley hypotheses, the descent and Eilenberg--Moore factorizations are equivalent whenever they exist. We thus obtain a stronger version of the Bénabou--Roubaud theorem for arbitrary augmented pseudocosimplicial objects, which we call the formal Bénabou--Roubaud theorem. Applied to indexed Čech objects associated with pseudofunctors A: C^op→ℋ, this theorem extends the classical Bénabou--Roubaud theorem to an arbitrary $2$-category ℋ.
ER  -

Abstract

Related

Cite