Cofibrantly generated lax orthogonal factorisation systems

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.