The Left Localization Principle, completions, and cofree $G$-spectra

TL;DR

This paper establishes conditions under which localizations preserve Quillen adjunctions and equivalences, and constructs a symmetric monoidal algebraic model for rational cofree G-spectra, linking localization, monoidal structures, and algebraic models.

Contribution

It provides new criteria for localizations to induce Quillen equivalences and constructs an algebraic model for rational cofree G-spectra using these principles.

Findings

Localization preserves Quillen adjunctions under mild conditions.

Conditions identified for localized adjunctions to be Quillen equivalences.

Constructed a symmetric monoidal algebraic model for rational cofree G-spectra.

Abstract

We show under mild hypotheses that a Quillen adjunction between stable model categories induces another Quillen adjunction between their left localizations, and we provide conditions under which the localized adjunction is a Quillen equivalence. Moreover, we show that in many cases the induced Quillen equivalence is symmetric monoidal. Using our results we construct a symmetric monoidal algebraic model for rational cofree $G$-spectra. In the process, we also show that $L$-complete modules provide an abelian model for derived complete modules.