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In Higher Topos Theory by J. Lurie, Def. 5.2.7.2, a localization functor between $\infty$-categories is defined as a functor having a fully faithful right adjoint.

Now, given any $\infty$-category $\mathscr C$, consider a fibrant replacement $\mathscr{C}\to \tilde{\mathscr{C}}$ in the model category of simplicial sets (Kan model structure). The replacement $\tilde{\mathscr C}$ is an infinity-groupoid, hence all of its morphisms are equivalences. So, is it true that we can regard $\tilde{\mathscr{C}}$ as a localization in the previous sense? If so, I have trouble in finding the right adjoint...

Thank you in advance for any help.

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A localization in the sense of Lurie is a reflective localization. In particular, the localized category must be a full subcategory of the original ∞-category. There are many ∞-categories C such that C̃ is not a full ∞-subcategory of C. For instance, take C to be the delooping of a monoid that is not a group.

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  • $\begingroup$ I see, you are right... so, the case of the fibrant replacement is also not a reflective localization? $\endgroup$
    – W. Rether
    Aug 11, 2019 at 15:52
  • $\begingroup$ What does "also" refer to? I was talking about C̃, which is the fibrant replacement of C. $\endgroup$
    – Dmitri P.
    Aug 11, 2019 at 16:10
  • $\begingroup$ Of course. I didn't remember that the delooping works as a fibrant replacement, sorry. $\endgroup$
    – W. Rether
    Aug 11, 2019 at 16:14

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