# nLab 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

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

$A\simeq \left[{C}^{\mathrm{op}},{\mathrm{sSet}}_{\mathrm{Quillen}}{\right]}_{\mathrm{proj},\mathrm{loc}}\stackrel{\stackrel{}{←}}{\underset{}{\to }}\left[{C}^{\mathrm{op}},{\mathrm{sSet}}_{\mathrm{Quillen}}{\right]}_{\mathrm{proj}}\phantom{\rule{thinmathspace}{0ex}}.$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

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

$D\stackrel{\stackrel{L}{←}}{\underset{}{↪}}{\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$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

category: reference

Revised on May 18, 2010 08:13:40 by Urs Schreiber (87.212.203.135)