Complexes of stable $\infty$-categories

TL;DR

This paper introduces the theory of complexes of stable $b$-categories, develops a totalization technique, and applies it to establish categorical dualities, interpret higher-dimensional perverse schobers, and propose a variant of homological mirror symmetry.

Contribution

It presents a new framework for categorical complexes, including a totalization method and applications to dualities, perverse schobers, and mirror symmetry.

Findings

Established a categorical Koszul duality generalizing derived Morita equivalences.

Interpreted spherical complexes as higher-dimensional perverse schobers.

Proposed and verified a variant of homological mirror symmetry for $\u211b ext{P}^2$.

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 $\mathbb{C}\mathrm{P}^2$.