Proxy-small objects present compactly generated categories

TL;DR

This paper establishes a new correspondence linking compactly generated triangulated categories with localizations of derived categories of ring spectra via proxy-small objects, offering insights into their structure and properties.

Contribution

It introduces a novel correspondence between presentations of categories and proxy-small objects, and characterizes proxy-smallness through coproduct preservation.

Findings

Established a correspondence between category presentations and proxy-small objects.

Provided a characterization of proxy-smallness via coproduct preservation.

Explored consequences of this correspondence in category theory.

Abstract

We develop a correspondence between presentations of compactly generated triangulated categories as localizations of derived categories of ring spectra and proxy-small objects, and explore some consequences. In addition, we give a characterization of proxy-smallness in terms of coproduct preservation of the associated corepresentable functor `up to base change'.