nLab Combinatorial model categories have presentations

This entry is about the article

• Daniel Dugger:

  
  

  Combinatorial model categories have presentations

  
  

  Adv. Math. 164 1 (2001) 177-201

  arXiv:math/0007068

  doi:10.1006/aima.2001.2015

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

• 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