This entry is about the article
Dan Dugger, Combinatorial model categories have presentations (arXiv), Adv. Math. 164 (2001), no. 1, 177-201
Abstract We show that every combinatorial model category can be obtained, up to Quillen equivalence, by localizing a model category of diagrams of simplicial sets. This says that any combinatorial model category can be built up from a category of “generators” and a set of “relations” — that is, any combinatorial model category has a presentation.
This builds on the companion paper
This shows that, up to Quillen equivalence, every combinatorial model category $A$ arises as the left Bousfield localization of the global projective model structure on simplicial presheaves $[C^{op}, sSet_{Quillen}]_{proj}$, for some small category $C$
Notice (as discussed at the relevant entries) that
every combinatorial model category can be enhanced to a combinatorial simplicial model category;
that the full enriched subcategories $A^\circ$ of these on fibrant-cofibrant objects are the locally presentable (∞,1)-categories;
that under this correspondence the global model structure on simplicial presheaves models the corresponding (∞,1)-category of (∞,1)-presheaves
Under this correspondence, Dugger’s theorem is precisely the model category-theoretic analog of the theorem that every locally presentable (∞,1)-category $D$ is a (∞,1)-categorical localization of an (∞,1)-category of (∞,1)-presheaves
This (∞,1)-category theortic interpretation of Dugger’s theorem is proposition A.3.7.6 in
Last revised on February 19, 2014 at 18:56:05. See the history of this page for a list of all contributions to it.