Small presentations of model categories and Vop\v{e}nka's principle

TL;DR

This paper extends the theory of small presentations in model categories, generalizing Dugger's theorem, and explores the role of Vopěnka's principle in establishing Quillen equivalences.

Contribution

It generalizes existing results on small presentations of model categories to broader classes, including those under Vopěnka's principle, and corrects previous inaccuracies.

Findings

Under Vopěnka's principle, certain model categories are Quillen equivalent to combinatorial ones.

Cofibrantly generated model categories with specific smallness conditions admit small presentations.

The paper corrects a previous mistake related to similar claims in the literature.

Abstract

We prove existence results for small presentations of model categories generalizing a theorem of D. Dugger from combinatorial model categories to more general model categories. Some of these results are shown under the assumption of Vop\v{e}nka's principle. Our main theorem applies in particular to cofibrantly generated model categories where the domains of the generating cofibrations satisfy a slightly stronger smallness condition. As a consequence, assuming Vop\v{e}nka's principle, such a cofibrantly generated model category is Quillen equivalent to a combinatorial model category. Moreover, if there are generating sets which consist of presentable objects, then the same conclusion holds without the assumption of Vop\v{e}nka's principle. We also correct a mistake from previous work that made similar claims.