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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

This shows that, up to Quillen equivalence, every combinatorial model category AA arises as the left Bousfield localization of the global projective model structure on simplicial presheaves [C op,sSet Quillen] proj[C^{op}, sSet_{Quillen}]_{proj}, for some small category CC

A[C op,sSet Quillen] proj,loc[C op,sSet Quillen] 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

Under this correspondence, Dugger’s theorem is precisely the model category-theoretic analog of the theorem that every locally presentable (∞,1)-category DD is a (∞,1)-categorical localization of an (∞,1)-category of (∞,1)-presheaves

DLPSh (,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

category: reference

Revised on February 19, 2014 18:56:05 by David Roberts (129.127.252.10)