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