Complexes of stable ∞-categories and higher Segal conditions

There exists an equivalence of (∞,2)-categories between the (∞,2)-category of complexes of stable ∞-categories and that of 2-simplicial stable ∞-categories, established by Dyckerhoff, which categorifies the classical Dold–Kan correspondence. It is well known that every simplicial abelian group is in particular a Kan complex, i.e. it admits certain horn fillers. In this talk, I will explain joint work with Dyckerhoff, Gödicke, and Walde, in which we show that every 2-simplicial stable ∞-category satisfies an analogous categorified Kan complex condition, and use this to characterize those 2-simplicial stable ∞-categories that satisfy the (higher) Segal conditions of Dyckerhoff–Kapranov.