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](http://arxiv.org/pdf/math/0210114.pdf) 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?



Verdier localization for stable $\infty$-categories
One good source is Blumberg-Gepner-Tabuada, "A Universal Characterization of Higher Algebraic K-Theory."


See Definition 5.4, which defines the Verdier quotient as the cofiber (In the oo-category of presentable stable oo-categories) of the fully faithful functor $\mathcal{C} \to \mathcal{T}$. 


Proposition 5.6 shows it can be characterized as a Bousfield localization, localizing at the morphisms whose cofibers are in the image of $\mathcal{C}$.


Proposition 5.9 deals with Adeel's comment: It shows $Ho(\mathcal{T})/Ho(\mathcal{C}) \simeq Ho(\mathcal{T}/\mathcal{C})$.


