On triangulated categories we have a notion of [Bridgeland stability conditions](http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf).


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 $hC$?



$\infty$-categorical understanding of Bridgeland stability?
Yes, this is chapter 7 of [Fosco Loregian's thesis](http://www.math.muni.cz/~loregianf/stuff/main.pdf), 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.


