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Combinatorial model categories have presentations

This entry is about the article

• Dan Dugger, Combinatorial model categories have presentations (arXiv)

  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 $[C^{\mathrm{op}},{\mathrm{sSet}}_{\mathrm{Quillen}}{]}_{\mathrm{proj}}$, for some small category $C$

$$A\simeq [C^{\mathrm{op}},{\mathrm{sSet}}_{\mathrm{Quillen}}{]}_{\mathrm{proj},\mathrm{loc}}\overleftarrow{\underset{}{\to }}[C^{\mathrm{op}},{\mathrm{sSet}}_{\mathrm{Quillen}}{]}_{\mathrm{proj}}.$$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

  $${\mathrm{PSh}}_{(\mathrm{\infty },1)}(C)\simeq A^{\circ }.$$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

$$D\stackrel{\stackrel{L}{\leftarrow }}{\underset{}{\hookrightarrow }}{\mathrm{PSh}}_{(\mathrm{\infty },1)}(C).$$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