Understanding the small object argument

TL;DR

This paper introduces an algebraic refinement of the small object argument, addressing its lack of universal property, convergence issues, and disconnection from other transfinite constructions in categorical algebra.

Contribution

It provides an algebraic version of the small object argument within natural weak factorisation systems, fixing key deficiencies.

Findings

The refined argument has a universal property.

It guarantees convergence of the construction.

It establishes connections with other transfinite categorical algebra methods.

Abstract

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.