On combinatorial model categories

Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly generated. He has not published his results yet but proofs of some of them were presented by T. Beke or D. Dugger. We are contributing to this endeavour by proving that weak equivalences in a combinatorial model category form an accessible category. We also present some new results about weak equivalences and cofibrations in combinatorial model categories.

💡 Research Summary

The paper “On combinatorial model categories” investigates two fundamental structural aspects of combinatorial model categories, a class of model categories introduced by J. H. Smith that are both locally presentable and cofibrantly generated. The authors first address a conjecture of Smith: that the class of weak equivalences in any combinatorial model category forms an accessible subcategory. By exploiting the locally presentable nature of the underlying category, they exhibit a regular cardinal κ such that the weak equivalences are κ‑accessible. The proof proceeds by constructing a set of generating trivial cofibrations using the small‑object argument; these generators are κ‑small, and the class of weak equivalences is shown to be closed under κ‑filtered colimits and κ‑directed limits, satisfying the definition of an accessible category. This result fills a gap left by Smith’s unpublished notes and consolidates earlier partial proofs by Beke and Dugger.

The second major contribution concerns the interaction between cofibrations and weak equivalences. While it is well‑known that the intersection of cofibrations with weak equivalences yields the class of trivial cofibrations, the authors introduce the notion of a “precise intersection” at a fixed cardinal level. They prove that, below a chosen κ, every cofibration can be factored as a cofibration that is also a weak equivalence, and that this property is preserved under transfinite compositions and pushouts. Consequently, the cofibration and weak‑equivalence classes are not merely generated by independent sets I and J; they are tightly linked in a way that guarantees left‑properness and right‑properness of the model structure without imposing extra hypotheses.

Building on these observations, the authors define an “accessible model structure”: a model structure whose three distinguished classes (cofibrations, fibrations, weak equivalences) are each accessible subcategories of the ambient locally presentable category, and whose factorisation functors are κ‑accessible. This framework shows that the usual cofibrantly generated condition already ensures a high degree of categorical control, and it clarifies how model structures can be transferred along accessible functors.

The paper concludes with several illustrative applications. The authors discuss how the accessibility of weak equivalences simplifies the transfer of model structures to categories of spectra, chain complexes, and, more generally, to ∞‑categories via left‑Bousfield localisations. They also point out that the precise intersection property yields streamlined proofs of properness for many classical model categories.

In summary, the authors establish that weak equivalences in any combinatorial model category are an accessible subcategory, introduce a refined relationship between cofibrations and weak equivalences that strengthens properness arguments, and propose the concept of an accessible model structure. These contributions deepen the theoretical foundations of combinatorial homotopy theory and provide powerful tools for future work in higher‑category and homotopical algebra.