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Dan Dugger proved that every combinatorial model category can be obtained up to Quillen equivalence as the localization of a model structure on simplicial presheaves on a small category $C$.

Is there a classification of those combinatorial model categories admitting a Dugger presentation where $C$ is Reedy, almost-Reedy, or generalized-Reedy?

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    $\begingroup$ Any small category is a localization of a direct (whence Reedy) category. Therefore, it follows straight away from Dugger’s theorem that any combinatorial model structure is Quillen equivalent to a localization of a Reedy model structure on simplicial presheaves on a small direct category. $\endgroup$
    – D.-C. Cisinski
    Commented Jul 22, 2018 at 21:01
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    $\begingroup$ @Denis-CharlesCisinski A reference please would be welcome. And I think that your comment is an answer. $\endgroup$
    – Philippe Gaucher
    Commented Jul 23, 2018 at 7:56
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    $\begingroup$ A possible reference consists to put together Cor. 7.3.10 and Prop. 7.1.12 from my lecture notes "Higher categories and homotopical algebra" (available from my webpage). But there are probably much older references. $\endgroup$
    – D.-C. Cisinski
    Commented Jul 23, 2018 at 9:29

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