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Combinatorial model categories have presentations

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

• Dan Dugger, Universal homotopy theories

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 $C$

$$A \simeq [C^{op}, sSet_{Quillen}]_{proj, loc} \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} [C^{op}, sSet_{Quillen}]_{proj} \,.$$

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

  $$PSh_{(\infty,1)}(C) \simeq A^{\circ} \,.$$

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

$$D \stackrel{\overset{L}{\leftarrow}}{\underset{}{\hookrightarrow}} PSh_{(\infty,1)}(C) \,.$$

This (∞,1)-category theortic interpretation of Dugger’s theorem is proposition A.3.7.6 in

• Jacob Lurie, Higher Topos Theory

category: reference