Verdier localization is one of the more intuitive ways to localize a triangulated category, "killing" a suitable class of objects via a functor which is universal with respect to this property.

I would like to know whether it is possible to reproduce the construction of $\mathcal{T}/\mathcal C$ in a $\infty$-stable setting. It seems a well-established folklore that the category of (a model for) stable $\infty$-categories is stable (!) under this sort of operation, but I can't find a reference in Lurie HA1.

In the DG model, there is a construction by Drinfeld which seems to do the job, but instead I would like to reproduce the fairly general construction of Neeman (*Triangulated Categories*), Ch. 2. Did anybody do this *naive* construction? Or rather there is a more conceptual approach?

Presenting a stable $\infty$-category via a stable model category, is Bousfield (which is, as far as I understand, only a *particular* case of Verdier) localization enough to cover "all" the interesting cases?

Higher algebra, Proposition 1.1.4.6) this is again a stable $\infty$-category. $\endgroup$