Cofibrantly generated lax orthogonal factorisation systems

TL;DR

This paper extends the theory of cofibrant generation of algebraic weak factorisation systems to broader contexts, introduces cofibrantly KZ-generated AWFSs as lax orthogonal, and compares two methods of constructing lax orthogonal AWFSs.

Contribution

It broadens the theory of cofibrant generation to non-locally presentable categories, defines cofibrantly KZ-generated AWFSs, and distinguishes between two construction methods for lax orthogonal AWFSs.

Findings

Cofibrantly KZ-generated AWFSs are always lax orthogonal.

The two known methods for building lax orthogonal AWFSs produce different systems.

Counterexample provided by continuous lattices shows limitations of cofibrant generation.

Abstract

The present note has three aims. First, to complement the theory of cofibrant generation of algebraic weak factorisation systems (AWFSs) to cover some important examples that are not locally presentable categories. Secondly, to prove that cofibrantly KZ-generated AWFSs (a notion we define) are always lax orthogonal. Thirdly, to show that the two known methods of building lax orthogonal AWFSs, namely cofibrantly KZ-generation and the method of "simple adjunctions", construct different AWFSs. We study in some detail the example of cofibrant KZ-generation that yields representable multicategories, and a counterexample to cofibrant generation provided by continuous lattices.