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