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Complexes of stable $\infty$-categories

Merlin Christ, Tobias Dyckerhoff, Tashi Walde

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

Complexes of stable $\infty$-categories

Abstract

We study complexes of stable -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 .
Paper Structure (48 sections, 46 theorems, 167 equations, 3 figures)

This paper contains 48 sections, 46 theorems, 167 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Rules of categorification
  3. Examples of categorical complexes
  4. The categorified Dold--Kan correspondence
  5. Categorical Koszul complexes and representation theory
  6. Categorical intersection complexes and lax algebraic geometry
  7. Fukaya-Seidel complexes in symplectic geometry
  8. Complexes of local systems on manifolds with corners
  9. Totalization constructions
  10. Mirror symmetry for categorical complexes
  11. Geometric structures associated with categorical complexes
  12. Calabi-Yau structures
  13. Spherical complexes and perverse schobers
  14. Lax additivity
  15. Relations to previous work
  16. ...and 33 more sections

Key Result

Lemma 2.2.3

We regard ${{\EuScript S}t_{k}}$ as an $\infty$-category (by discarding noninvertible $2$-morphisms) and consider further the $\infty$-categories The restriction functors are equivalences of $\infty$-categories.

Figures (3)

  • Figure 2.4.1: The four lax cones with base given by a functor $F\colon \EuScript{A} \to \EuScript{B}$.
  • Figure 7.3.1: Decomposition of $\Gamma$.
  • Figure 7.3.2: Diagram computing the topological Fukaya category of $\Gamma$.

Theorems & Definitions (139)

  • Definition 2.2.1
  • Lemma 2.2.3
  • proof : Proof sketch
  • Definition 2.2.5
  • Definition 2.2.6
  • Theorem 2.4.2: Lax Additivity, CDW24
  • Remark 2.4.3
  • Remark 2.4.4
  • Definition 3.1.1
  • Remark 3.1.2
  • ...and 129 more