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There are many models for $\infty$-categories. One of them is relative categories – AKA categories with weak equivalences – which have a model structure due to Barwick–Kan. Another one is simplicial categories with Bergner's structure. It is otherwise known that these two are Quillen equivalent, but the equivalence goes through complete Segal spaces and quasi-categories (see e.g. Bergner's book The Homotopy Theory of $(\infty,1)$-categories for a recap).

There is a functor from relative categories to simplicial categories given by simplicial localizations. This is a very naïve question, but is it part of a Quillen equivalence? (Honestly, I expect that the answer is no, otherwise it would probably have been mentioned in the reference books I've read, but you never know...) If not, is it possible to find a direct one between these two model categories?

(I'm aware that the answer to my title taken literally could be "yes, define a model structure on simplicial categories through transfer, and voilà". That's not really what I'm expecting. The model structure on both is fixed.)

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  • $\begingroup$ Maybe a more precise way of formulating what you want is whether the simplicial localization induces an equivalence of relative categories (I assume you are not too committed to keep the cofibrations or the fibrations fixed :)) $\endgroup$
    – Denis Nardin
    Commented Sep 25, 2019 at 9:42
  • $\begingroup$ @DenisNardin Sure, that works too! $\endgroup$
    – Najib Idrissi
    Commented Sep 25, 2019 at 9:46
  • $\begingroup$ In this case I know that a Quillen equivalence between two model categories induces a Dwyer–Kan equivalence (i.e. an equivalence in Bergner's structure) between the hammock localizations. (First taking the subcategories of fibrant-cofibrant objects.) But I don't know if it's true for any weak equivalence of relative categories. $\endgroup$
    – Najib Idrissi
    Commented Sep 25, 2019 at 14:12
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    $\begingroup$ Is there a simplicial localization functor which is adjoint (either left or right)? $\endgroup$
    – Valery Isaev
    Commented Sep 26, 2019 at 20:14
  • $\begingroup$ @ValeryIsaev I don't know, is there? That's part of my question. $\endgroup$
    – Najib Idrissi
    Commented Sep 27, 2019 at 6:35

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