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 wellbehaved in the presence of BeckChevalley 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 higherdimensional perverse schobers, and introduce CalabiYau structures on categorical complexes to capture noncommutative orientation data. A variant of homological mirror symmetry for categorical complexes is proposed and verified for $\mathbb{C}\mathrm{P}^2$. Finally, we develop the concept of a lax additive $(\infty,2)$category and propose it as a suitable framework to formulate further aspects of categorified homological algebra.
 Publication:

arXiv eprints
 Pub Date:
 January 2023
 DOI:
 10.48550/arXiv.2301.02606
 arXiv:
 arXiv:2301.02606
 Bibcode:
 2023arXiv230102606C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Category Theory;
 Mathematics  KTheory and Homology;
 18N60;
 18E05;
 18N25;
 53D37;
 18G35
 EPrint:
 139 pages