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Integration of Hochschild cohomology, derived Picard groups and uniqueness of lifts

Sebastian Opper

TL;DR

This work develops a general integration map from the first Hochschild cohomology $\mathrm{HH}^1_+(\mathbb{A},\mathbb{A})$ of a cohomologically unital $A_\infty$-category over a characteristic-zero field to its derived Picard group, via a complete pre-Lie exponential built from the brace algebra structure on the Hochschild space. The map $\exp_{\mathbb{A}}$ is shown to be injective under natural conditions, with naturality under quasi-equivalences, and its image captures higher “identity-component” information beyond homotopy categories. The authors develop a robust integration framework for pre-Lie and brace algebras, relate $\infty$-isotopies to Maurer–Cartan elements, and connect these to lifts of functors, thereby yielding a new obstruction-theoretic criterion for uniqueness of lifts. They apply the theory to derived Picard groups of Fukaya categories, including wrapped and compact versions of cotangent bundles and their plumbings, and discuss extensions to partially wrapped Fukaya categories following Haiden–Katzarkov–Kontsevich. The results provide explicit tools to access parts of the identity component of $\mathcal{D}\mathrm{Pic}(\mathbb{A})$ from Hochschild data, with implications for Koszul duality, mapping-class-group actions, and deformation theory in symplectic geometry and beyond.

Abstract

The paper introduces a partial integration map from the first Hochschild cohomology of any cohomologically unital A-infinity category over a field of characteristic zero to its derived Picard group. We discuss useful properties such as injectivity, naturality and the relation with the Baker-Campbell-Hausdorff formula. Based on the image of the integration map we propose a candidate for the identity component of the derived Picard group in the case of finite-dimensional graded algebras. As a first application of the integration map it is shown that the vanishing of its domain is a necessary condition for the uniqueness of lifts of equivalences from the homotopy category to the A-infinity-level. The final part contains applications to derived Picard groups of wrapped and compact Fukaya categories of cotangent bundles and their plumbings and an outlook on applications to derived Picard groups of partially wrapped Fukaya categories after Haiden-Katzarkov-Kontsevich.

Integration of Hochschild cohomology, derived Picard groups and uniqueness of lifts

TL;DR

This work develops a general integration map from the first Hochschild cohomology of a cohomologically unital -category over a characteristic-zero field to its derived Picard group, via a complete pre-Lie exponential built from the brace algebra structure on the Hochschild space. The map is shown to be injective under natural conditions, with naturality under quasi-equivalences, and its image captures higher “identity-component” information beyond homotopy categories. The authors develop a robust integration framework for pre-Lie and brace algebras, relate -isotopies to Maurer–Cartan elements, and connect these to lifts of functors, thereby yielding a new obstruction-theoretic criterion for uniqueness of lifts. They apply the theory to derived Picard groups of Fukaya categories, including wrapped and compact versions of cotangent bundles and their plumbings, and discuss extensions to partially wrapped Fukaya categories following Haiden–Katzarkov–Kontsevich. The results provide explicit tools to access parts of the identity component of from Hochschild data, with implications for Koszul duality, mapping-class-group actions, and deformation theory in symplectic geometry and beyond.

Abstract

The paper introduces a partial integration map from the first Hochschild cohomology of any cohomologically unital A-infinity category over a field of characteristic zero to its derived Picard group. We discuss useful properties such as injectivity, naturality and the relation with the Baker-Campbell-Hausdorff formula. Based on the image of the integration map we propose a candidate for the identity component of the derived Picard group in the case of finite-dimensional graded algebras. As a first application of the integration map it is shown that the vanishing of its domain is a necessary condition for the uniqueness of lifts of equivalences from the homotopy category to the A-infinity-level. The final part contains applications to derived Picard groups of wrapped and compact Fukaya categories of cotangent bundles and their plumbings and an outlook on applications to derived Picard groups of partially wrapped Fukaya categories after Haiden-Katzarkov-Kontsevich.
Paper Structure (56 sections, 54 theorems, 118 equations)

This paper contains 56 sections, 54 theorems, 118 equations.

Table of Contents

  1. Pre-Lie algebras, brace algebras and symmetric brace algebras
  2. Pre-Lie algebras
  3. Example: the Hochschild space of a vector space
  4. Brace algebras and symmetric brace algebras
  5. Graphs, cocategories and Hochschild spaces
  6. Graphs and cocategories
  7. Tensor cocategories
  8. The differential graded case
  9. The Hochschild space of a $\Bbbk$-graph
  10. The brace algebra structure on the Hochschild space
  11. Restricted naturality
  12. Reminder on A-infinity categories
  13. $A_\infty$-categories via Maurer-Cartan elements in Hochschild spaces
  14. Homotopy categories and $c$-unitality
  15. $A_\infty$-functors and functor categories
  16. ...and 41 more sections

Key Result

Theorem A

Let $\mathbb{A}$ be a cohomologically unital $A_\infty$-category over a field $\Bbbk$ of characteristic $0$. There exists a group homomorphism \begin{tikzcd} \exp_{\bA}: \HHH^1_+(\bA, \bA) \arrow{r} & \DPic(\bA). \end{tikzcd}Here, $\operatorname{HH}^1_+(\mathbb{A}, \mathbb{A})$ denotes the classes

Theorems & Definitions (131)

  • Theorem A: \ref{['TheoremNaturalityExponentialQuasiIsos']} & \ref{['CorollaryExponentialInjectiveMainTheorem']}
  • Theorem B: \ref{['TheoremNaturalityExponentialQuasiIsos']}
  • Theorem C: \ref{['TheoremUniquenessLiftingVanishing']}
  • Theorem D: \ref{['CorollaryTheoremE']}
  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: Universal property
  • ...and 121 more