Small presentations of model categories and Vopěnka’s principle

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 Vopenka'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 Vopenka'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 Vopenka's principle. We also correct a mistake from previous work that made similar claims.