On solid and rigid monoids in monoidal categories

TL;DR

This paper introduces solid and rigid monoids in monoidal categories, exploring their properties, relationships with localizations, and applications to ring spectra, including a characterization of connective solid ring spectra.

Contribution

It defines solid and rigid monoids, establishes their correspondence with localizations, and applies these concepts to classify certain ring spectra in stable homotopy theory.

Findings

Solid monoids correspond to smashing localizations and colocalizations.

Rigid monoids are characterized as localizations of the monoidal unit.

Connective solid ring spectra are Moore spectra of subrings of the rationals.

Abstract

We introduce the notion of solid monoid and rigid monoid in monoidal categories and study the formal properties of these objects in this framework. We show that there is a one to one correspondence between solid monoids, smashing localizations and mapping colocalizations, and prove that rigid monoids appear as localizations of the unit of the monoidal structure. As an application, we study solid and rigid ring spectra in the stable homotopy category and characterize connective solid ring spectra as Moore spectra of subrings of the rationals.