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 $\mathcal H$ be a $2$-category, and let $\operatorname{suc}=\mathsf{1}+(-)$ be the successor monad on the augmented simplex category. Precomposition with $\operatorname{suc}$ induces the décalage pseudomonad $\mathbb S$ on the $2$-category of augmented pseudocosimplicial objects with values in $\mathcal H$. 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 $A\colon C^{\mathrm{op}}\to\mathcal H$, where $\mathcal H$ is an arbitrary $2$-category. Consequently, every $\mathbb S$-pseudoalgebra is of absolute descent. In particular, this holds for every free $\mathbb S$-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 $A\colon C^{\mathrm{op}}\to\mathcal H$, this theorem extends the classical Bénabou--Roubaud theorem to an arbitrary $2$-category $\mathcal H$.