The algebraic small object argument as a saturation

TL;DR

This paper explores the structure of left maps in algebraic weak factorization systems, showing they can be built from generators using classical cell-complex operations, and introduces saturation principles for property extension.

Contribution

It characterizes left maps in algebraic weak factorization systems as generated by specific operations and formulates saturation principles for extending properties from generators to all left maps.

Findings

Left maps can be constructed from generators using classical cell-complex operations.

Saturation principles describe closure conditions for property extension.

Analysis of free monad construction on pointed endofunctors underpins the results.

Abstract

We analyze the structure of left maps in algebraic weak factorization systems constructed using Garner's algebraic small object argument. We find that any left map can be constructed from generators in Bourke and Garner's double category of left maps by operations that parallel the classical cell-complex-forming operations of Quillen's small object argument (coproducts, cobase changes, transfinite composites, and retracts). Our main theorems are phrased as "saturation" principles, which express the closure conditions necessary for a given property or structure to extend from generators to all left maps. The core of the argument is an analysis of the construction of the free monad on a pointed endofunctor.