On the Cofibrant Generation of Model Categories

The paper studies the problem of the cofibrant generation of a model category. We prove that, assuming Vop\v{e}nka’s principle, every cofibrantly generated model category is Quillen equivalent to a combinatorial model category. We discuss cases where this result implies that the class of weak equivalences in a cofibrantly generated model category is accessibly embedded. We also prove a necessary condition for a model category to be cofibrantly generated by a set of generating cofibrations between cofibrant objects.

💡 Research Summary

The paper addresses a central structural question in the theory of model categories: under what conditions can a cofibrantly generated model category be replaced, up to Quillen equivalence, by a combinatorial one? The authors work under the set‑theoretic hypothesis known as Vopěnka’s principle, a strong large‑cardinal axiom that guarantees the existence of certain reflective subcategories and the accessibility of classes defined by a set of morphisms.

The first major result shows that any cofibrantly generated model category (\mathcal{M}) admits a Quillen equivalence to a combinatorial model category (\mathcal{K}). The construction proceeds by selecting a regular cardinal (\lambda) large enough that the generating cofibrations and trivial cofibrations of (\mathcal{M}) are (\lambda)-presentable. Using Vopěnka’s principle, the authors produce a small (\lambda)-accessible subcategory (\mathcal{K}) together with an adjunction \