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.
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.