Complexes of stable $\infty$-categories

Abstract

We study complexes of stable $\infty$-categories, referred to as categorical complexes. As we demonstrate, examples of such complexes arise in a variety of subjects including representation theory, algebraic geometry, symplectic geometry, and differential topology. One of the key techniques we introduce is a totalization construction for categorical cubes which is particularly well-behaved in the presence of Beck-Chevalley conditions. As a direct application we establish a categorical Koszul duality result which generalizes previously known derived Morita equivalences among higher Auslander algebras and puts them into a conceptual context. We explain how spherical categorical complexes can be interpreted as higher-dimensional perverse schobers, and introduce Calabi-Yau structures on categorical complexes to capture noncommutative orientation data. A variant of homological mirror symmetry for categorical complexes is proposed and verified for $\mathbb{C}\mathrm{P}^2$.