On a fat small object argument

TL;DR

This paper explores how good colimits, a concept introduced by J. Lurie, relate to transfinite composites and demonstrate their role in simplifying the small object argument by removing the need for large objects.

Contribution

It establishes a connection between good colimits and transfinite composites, showing how they can replace large objects in the small object argument.

Findings

Good colimits generalize transfinite composites.

They eliminate the need for large objects in the small object argument.

Enhance understanding of cofibrant generation in locally presentable categories.

Abstract

Good colimits introduced by J. Lurie generalize transfinite composites and provide an important tool for understanding cofibrant generation in locally presentable categories. We will explore the relation of good colimits to transfinite composites further and show, in particular, how they eliminate the use of large objects in the usual small object argument.