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Relative Calabi-Yau structures and perverse schobers on surfaces

Merlin Christ

Abstract

We give a treatment of relative Calabi--Yau structures on functors between $R$-linear stable $\infty$-categories, with $R$ any $\mathbb{E}_\infty$-ring spectrum, generalizing previous treatments in the setting of dg-categories. Using their gluing properties, we further construct relative Calabi--Yau structures on the global sections of perverse schobers, i.e. categorified perverse sheaves, on surfaces with boundary. We treat examples related to Fukaya categories and representation theory. In a related direction, we define the monodromy of a perverse schober parametrized by a ribbon graph on a framed surface and show that it forms a local system of stable $\infty$-categories.

Relative Calabi-Yau structures and perverse schobers on surfaces

Abstract

We give a treatment of relative Calabi--Yau structures on functors between -linear stable -categories, with any -ring spectrum, generalizing previous treatments in the setting of dg-categories. Using their gluing properties, we further construct relative Calabi--Yau structures on the global sections of perverse schobers, i.e. categorified perverse sheaves, on surfaces with boundary. We treat examples related to Fukaya categories and representation theory. In a related direction, we define the monodromy of a perverse schober parametrized by a ribbon graph on a framed surface and show that it forms a local system of stable -categories.
Paper Structure (30 sections, 56 theorems, 182 equations)

This paper contains 30 sections, 56 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Relative Calabi--Yau structures
  3. Perverse schobers and relative Calabi--Yau structures
  4. Examples: Fukaya categories and Fukaya-type categories
  5. Acknowledgments
  6. Notation
  7. Linear infinity-categories and Hochschild homology
  8. Linear infinity-categories
  9. Dualizable infinity-categories
  10. Duals of bimodules
  11. Smooth and proper linear infinity-categories
  12. Traces
  13. Hochschild homology
  14. Relative Calabi--Yau structures
  15. Left Calabi--Yau structures
  16. ...and 15 more sections

Key Result

Theorem 1

If the functors $\mathcal{B}_1\times \mathcal{B}_2\to \mathcal{C}_1$ and $\mathcal{B}_2\times \mathcal{B}_3\to \mathcal{C}_2$ carry $R$-linear left $n$-Calabi--Yau structures, which are compatible at $\mathcal{B}_2$, then the functor $\mathcal{B}_1\times \mathcal{B}_3\to \mathcal{D}$ inherits an $R$

Theorems & Definitions (155)

  • Theorem 1: \ref{['thm:leftCYglue']},BD19 for $R=k$ a field
  • Theorem 2: \ref{['thm:rightCYglue']}
  • Proposition 1: \ref{['prop:schobersfrommonodromy']}
  • Proposition 2: Combine \ref{['prop:localmodel', 'prop:loccy']}
  • Theorem 3: \ref{['thm:FukayaCY']}
  • Theorem 4: \ref{['thm:FSschober']}
  • Definition 2.1
  • Definition 2.2: $\!\!$HA
  • Remark 2.3
  • Lemma 2.4
  • ...and 145 more