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A proof of Kontsevich-Soibelman conjecture

Alexander I. Efimov

TL;DR

The paper proves the Kontsevich–Soibelman conjecture for essentially small $A_{ ablafty}$-(pre-)categories over a field, establishing a bijection between their quasi-equivalence classes and those of essentially small $A_{ ablafty}$-categories with identities. It introduces minimal models and Hochschild cohomology for graded pre-categories, proves a Main Lemma ensuring invariance of Hochschild cohomology under graded-pre-category equivalences, and develops obstruction theory to transfer $A_{ ablafty}$-structures across equivalences. A robust theory of twisted complexes over $A_{ ablafty}$-pre-categories is developed, yielding a quasi-equivalence-invariant pre-triangulated envelope and aligning with the classical twisted complexes for ordinary $A_{ ablafty}$-categories. The results justify replacing Fukaya $A_{ ablafty}$-pre-categories with quasi-equivalent actual $A_{ ablafty}$-categories, with implications for homological mirror symmetry and categorical constructions in symplectic geometry.

Abstract

It is well known that "Fukaya category" is in fact an $A_{\infty}$-pre-category in sense of Kontsevich and Soibelman \cite{KS}. The reason is that in general the morphism spaces are defined only for transversal pairs of Lagrangians, and higher products are defined only for transversal sequences of Lagrangians. In \cite{KS} it is conjectured that for any graded commutative ring $k,$ quasi-equivalence classes of $A_{\infty}$-pre-categories over $k$ are in bijection with quasi-equivalence classes of $A_{\infty}$-categories over $k$ with strict (or weak) identity morphisms. In this paper we prove this conjecture for essentially small $A_{\infty}$-(pre-)categories, in the case when $k$ is a field. In particular, it follows that we can replace Fukaya $A_{\infty}$-pre-category with a quasi-equivalent actual $A_{\infty}$-category. We also present natural construction of pre-triangulated envelope in the framework of $A_{\infty}$-pre-categories. We prove its invariance under quasi-equivalences.

A proof of Kontsevich-Soibelman conjecture

TL;DR

The paper proves the Kontsevich–Soibelman conjecture for essentially small -(pre-)categories over a field, establishing a bijection between their quasi-equivalence classes and those of essentially small -categories with identities. It introduces minimal models and Hochschild cohomology for graded pre-categories, proves a Main Lemma ensuring invariance of Hochschild cohomology under graded-pre-category equivalences, and develops obstruction theory to transfer -structures across equivalences. A robust theory of twisted complexes over -pre-categories is developed, yielding a quasi-equivalence-invariant pre-triangulated envelope and aligning with the classical twisted complexes for ordinary -categories. The results justify replacing Fukaya -pre-categories with quasi-equivalent actual -categories, with implications for homological mirror symmetry and categorical constructions in symplectic geometry.

Abstract

It is well known that "Fukaya category" is in fact an -pre-category in sense of Kontsevich and Soibelman \cite{KS}. The reason is that in general the morphism spaces are defined only for transversal pairs of Lagrangians, and higher products are defined only for transversal sequences of Lagrangians. In \cite{KS} it is conjectured that for any graded commutative ring quasi-equivalence classes of -pre-categories over are in bijection with quasi-equivalence classes of -categories over with strict (or weak) identity morphisms. In this paper we prove this conjecture for essentially small -(pre-)categories, in the case when is a field. In particular, it follows that we can replace Fukaya -pre-category with a quasi-equivalent actual -category. We also present natural construction of pre-triangulated envelope in the framework of -pre-categories. We prove its invariance under quasi-equivalences.

Paper Structure

This paper contains 14 sections, 19 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on $A_{\infty}$-(pre-)categories
  3. Non-unital $A_{\infty}$-algebras and $A_{\infty}$-categories
  4. Identity morphisms
  5. $A_{\infty}$-pre-categories
  6. Main Theorem
  7. From essentially small to small
  8. Minimal models
  9. Hochshild cohomology of graded pre-categories
  10. Main Lemma
  11. $A_{\infty}\text{-}$structures on a graded pre-category
  12. Invariance Theorem
  13. Proof of Main Theorem
  14. Twisted complexes over $A_{\infty}\text{-}$pre-categories

Key Result

Theorem 1.2

Let $k$ be a field. Then quasi-equivalence classes of essentially small $A_{\infty}\text{-}$pre-categories over $k$ are in bijection with quasi-equivalence classes of essentially small $A_{\infty}\text{-}$categories over $k$ with strict (or weak) identity morphisms.

Theorems & Definitions (59)

  • Conjecture 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 49 more