Localizations and completions of stable $\infty$-categories

TL;DR

This paper generalizes classical homology localization and nilpotent completion results to a broad class of stable $$-categories with a compatible $t$-structure, establishing conditions under which various completions and localizations coincide.

Contribution

It extends Bousfield's results to presentably symmetric monoidal stable $$-categories with a multiplicative left-complete $t$-structure, clarifying when different types of completions agree.

Findings

$E$-nilpotent completion, $E$-localization, and formal completion coincide on bounded below objects under certain conditions.

Provides a framework for understanding localizations and completions in a broad categorical setting.

Generalizes classical homology localization results to $$-categories.

Abstract

We extend some classical results of Bousfield on homology localizations and nilpotent completions to a presentably symmetric monoidal stable $\infty$-category $\mathscr{M}$ admitting a multiplicative left-complete $t$-structure. If $E$ is a homotopy commutative algebra in $\mathscr{M}$ we show that $E$-nilpotent completion, $E$-localization, and a suitable formal completion agree on bounded below objects when $E$ satisfies some reasonable conditions.