Monadic resolutions for generalized spaces

TL;DR

This paper generalizes monadic resolutions from classical spaces to $$-topoi, applying to equivariant and motivic homotopy theories, and proves convergence of key spectral sequences in these contexts.

Contribution

It extends monadic resolution techniques to $$-topoi and demonstrates their application to convergence results in equivariant and motivic homotopy theories.

Findings

Proves a principal fibration lemma in $$-topoi.

Shows convergence of the unstable Adams--Novikov spectral sequence over algebraically closed fields.

Extends monadic resolutions to new settings like equivariant and motivic spaces.

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.