# Verdier localization for stable $\infty$-categories

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?

• In the DG setting, the DG quotient constructions of Keller and Drinfel'd are models for the homotopy cofibre (in the Morita model structure on the category of small DG categories). This suggests defining the Verdier localization of a stable $\infty$-category by a stable sub-$\infty$-category as the cofibre of the inclusion. By (Higher algebra, Proposition 1.1.4.6) this is again a stable $\infty$-category. – AAK Jun 22 '14 at 3:41
• By "cofibre of the inclusion" do you mean the (homotopy) pushout of $* \leftarrow \mathcal C \hookrightarrow \mathcal T$? Extremely nice characterization! I'll be happy to accept as an answer a full explanation. Thank you! – Fosco Jun 22 '14 at 10:07
• Right. However I do not how to prove that $\mathrm{ho}(\mathcal{T}/\mathcal{C}) = \mathrm{ho}(\mathcal{T})/\mathrm{ho}(\mathcal{C})$, i.e. the homotopy category of the $\infty$-Verdier localization is the Verdier localization of the homotopy categories. – AAK Jun 22 '14 at 16:50
• @tetrapharmakon as you point out, Bousfield localization is Verdier localization with adjoints, which is a very particular case. It would be very nice to get a generalization of Verdier localization in general, but I don't even know if such a thing exists at the level of model categories. Maybe this is a question to be considered before. – Fernando Muro Jun 23 '14 at 10:29

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 $Ho(\mathcal{T})/Ho(\mathcal{C}) \simeq Ho(\mathcal{T}/\mathcal{C})$.