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Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I

Maxim Kontsevich, Yan Soibelman

TL;DR

The notes construct a geometric dictionary linking A$_\infty$-algebras and categories with non-commutative dg-manifolds, using coalgebras to model spaces, and develop Hochschild (co)chain theories, smoothness, compactness, and Calabi–Yau structures within this framework. They extend Deligne-type formality to a cylinder-based colored operad, connect to 2D TFTs, and formulate degeneration of Hodge-to-de Rham in the non-commutative setting, positing links to NC motives and moduli of A$_\infty$-categories. The work provides a coherent geometric toolkit for non-commutative spaces, including morphisms, inner Hom, and functorial constructions, and sketches a program toward a non-commutative Hodge theory and motivic perspective. Overall, it advances a robust framework for translating A$_\infty$-theory into non-commutative geometry with deep connections to Calabi–Yau structures and topological field theories.

Abstract

We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne's conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product. This action is essentially equivalent to the structure of 2-dimensional Topological Field Theory associated with a Calabi-Yau category.

Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I

TL;DR

The notes construct a geometric dictionary linking A-algebras and categories with non-commutative dg-manifolds, using coalgebras to model spaces, and develop Hochschild (co)chain theories, smoothness, compactness, and Calabi–Yau structures within this framework. They extend Deligne-type formality to a cylinder-based colored operad, connect to 2D TFTs, and formulate degeneration of Hodge-to-de Rham in the non-commutative setting, positing links to NC motives and moduli of A-categories. The work provides a coherent geometric toolkit for non-commutative spaces, including morphisms, inner Hom, and functorial constructions, and sketches a program toward a non-commutative Hodge theory and motivic perspective. Overall, it advances a robust framework for translating A-theory into non-commutative geometry with deep connections to Calabi–Yau structures and topological field theories.

Abstract

We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne's conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product. This action is essentially equivalent to the structure of 2-dimensional Topological Field Theory associated with a Calabi-Yau category.

Paper Structure

This paper contains 54 sections, 38 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. $A_{\infty}$-algebras as spaces
  3. Some applications of the geometric language
  4. Contents of the paper
  5. Generalization to $A_{\infty}$-categories
  6. $A_{\infty}$-algebras and non-commutative dg-manifolds
  7. Coalgebras and non-commutative schemes
  8. Coalgebras as functors
  9. Smooth thin schemes
  10. Inner Hom
  11. $A_{\infty}$-algebras
  12. Main definitions
  13. Minimal models of $A_{\infty}$-algebras
  14. Centralizer of an $A_{\infty}$-morphism
  15. Non-commutative dg-line ${\bf L}$ and weak unit
  16. ...and 39 more sections

Key Result

Theorem 2.1.1

Let $F : Alg_{{\cal C}^f} \to Sets$ be a covariant functor commuting with finite projective limits. Then it is isomorphic to a functor of the type $A \mapsto Hom_{Coalg_{{\cal C}}} ( A^{\ast}, B )$ for some counital coalgebra $B$. Moreover, the category of such functors is equivalent to the category

Theorems & Definitions (111)

  • Theorem 2.1.1
  • Proposition 2.1.2
  • Example 2.1.3
  • Definition 2.1.4
  • Definition 2.1.5
  • Example 2.1.6
  • Example 2.1.7
  • Example 2.1.8
  • Example 2.1.9
  • Remark 2.1.10
  • ...and 101 more