Recognition of commutative algebra spectra through an idempotent quasiadjunction

TL;DR

This paper establishes a recognition principle for certain algebraic spectra using advanced operadic and model theoretical tools, extending classical results to a more general setting.

Contribution

It introduces a new recognition principle for commutative algebra spectra via idempotent quasiadjunctions and relative operads, broadening the scope of classical methods.

Findings

Proved a recognition principle for $$-loop pairs of spaces of connective commutative algebra spectra.

Generalized classical recognition principles using relative operads.

Applied idempotent quasiadjunctions to handle model theoretical aspects.

Abstract

In this article a recognition principle for $\infty$-loop pairs of spaces of connective commutative algebra spectra over connective commutative ring spectra is proved. This is done by generalizing the classical recognition principle for connective commutative ring spectra using relative operads. The machinery of idempotent quasiadjunctions is used to handle the model theoretical aspects of the proof.