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.

中文翻译：

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共纤维生成的松散正交分解系统

本说明有三个目的。首先，补充代数弱分解系统 (awfss) 的共纤维生成理论，以涵盖一些不是局部可表示类别的重要示例。其次，要证明 cofibrantly kz 生成的 awfss（我们定义的一个概念）总是松散的正交。第三，为了证明构建松散正交 awfss 的两种已知方法，即共纤 kz 生成和“简单附加”方法，构建了不同的 awfss。我们详细研究了产生可表示的多类别的共纤维 kz 生成的例子，以及由连续晶格提供的共纤维生成的反例。