Monadic resolutions for generalized spaces
Tom Bachmann, Anton Engelmann, Klaus Mattis
TL;DR
The paper develops a unified framework to generalize Bousfield–Kan monadic resolutions to ∞-topoi and pro-objects, introducing a cobar/monadic completion CB(X) and a pro-object tower T^• to study L-localization and completion in broad contexts. It proves a generalized principal fibration lemma ensuring convergence of monadic resolutions for suitable fiber sequences, provided axioms C, M, S hold; this underpins convergence results across spaces, sheaves, equivariant, and motivic spaces. In concrete settings, the authors identify unstable localization/completion functors for various A (e.g. π_0(A) = S^{-1}ℤ or ℤ/n, Milnor–Witt/A, etc.), and derive unstable Adams spectral sequences via the monadic cobar construction, including specialized results for Burnside and Milnor–Witt resolutions in equivariant and motivic homotopy theory. The framework yields computable Tot- or totalization formulas, enabling explicit convergence statements and spectral sequence computations in unstable equivariant and motivic homotopy theories, with particular emphasis on nilpotent, connected objects and fields with finite 2-etale cohomological dimension. Overall, the work provides a powerful, modular toolbox for localization and completion in generalized spaces, with concrete applications to genuine G-spaces and motivic spaces over perfect fields.
Abstract
We extend the work of Bousfield and Kan on monadic resolutions of spaces to $\infty$-topoi, with applications to genuine $G$-equivariant spaces ($G$ a finite group) and motivic spaces over a perfect field. In particular, we give a proof of the principal fibration lemma in this context. We apply the principal fibration lemma to prove convergence of several kinds of monadic resolutions in unstable equivariant and motivic homotopy theory. For example, we show that, over an algebraically closed field, the unstable Adams--Novikov spectral sequence (i.e., the monadic resolution corresponding to the algebraic cobordism spectrum $\mathrm{MGL}$) converges for all nilpotent, connected, $2$-effective motivic spaces.