TITLE: $\infty$-categorical understanding of Bridgeland stability?
QUESTION [8 upvotes]: On triangulated categories we have a notion of Bridgeland stability conditions.
Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become Bridgeland stability conditions after passing to the homotopy category $hC$?

REPLY [6 votes]: Yes, this is chapter 7 of Fosco Loregian's thesis, linked from his webpage. The paper Simone Virili linked to is one of 3 papers making up the thesis. Specifically, Section 7.2.1 discusses the topology, after Bridgeland, and proves the comparison you asked for. The setting is a slight generalization of that of Bridgeland's original papers, as Loregian makes clear.