This entry is about the article
Combinatorial model categories have presentations
Adv. Math. 164 1 (2001) 177-201
on combinatorial model categories and their identifications with Bousfield localizations of model categories of simplicial presheaves.
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
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