Homology, Homotopy and Applications

Volume 20 (2018)

Number 1

Small presentations of model categories and Vopěnka’s principle

Pages: 303 – 328

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n1.a18


G. Raptis (Fakultät für Mathematik, Universität Regensburg, Germany)

J. Rosický (Department of Mathematics and Statistics, Faculty of Sciences, Masaryk University, Brno, Czech Republic)


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.


cofibrantly generated model category, combinatorial model category, simplicial presheaves, Vopěnka’s principle

2010 Mathematics Subject Classification

18C35, 18G55, 55U35

Received 11 July 2017

Received revised 10 October 2017

Published 28 February 2018