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Mat. Sb., 2011, Volume 202, Number 4, Pages 65–84 (Mi msb7753)  

This article is cited in 1 scientific paper (total in 1 paper)

A proof of the Kontsevich-Soǐbel'man conjecture

A. I. Efimov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: It is well known that the ‘Fukaya category’ is actually an $A_\infty$-precategory in the sense of Kontsevich and Soǐbel'man. This is related to the fact that, generally speaking, the morphism spaces are defined only for transversal pairs of Lagrangian submanifolds, and higher multiplications are defined only for transversal sequences of Lagrangian submanifolds. Kontsevich and Soǐbel'man made the following conjecture: for any graded commutative ring $k$, the quasi-equivalence classes of $A_\infty$-precategories over $k$ are in bijection with the quasi-equivalence classes of $A_\infty$-categories over $k$ with strict (or weak) identity morphisms.
In this paper this conjecture is proved for essentially small $A_\infty$-(pre)categories when $k$ is a field. In particular, this implies that the Fukaya $A_\infty$-precategory can be replaced with a quasi-equivalent actual $A_\infty$-category.
Furthermore, a natural construction of the pretriangulated envelope for $A_\infty$-precategories is presented and it is proved that it is invariant under quasi-equivalences.
Bibliography: 8 titles.

Keywords: $A_\infty$-categories, Fukaya category, homological mirror symmetry.

DOI: https://doi.org/10.4213/sm7753

Full text: PDF file (583 kB)
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English version:
Sbornik: Mathematics, 2011, 202:4, 527–546

Bibliographic databases:

Document Type: Article
UDC: 512.66
MSC: Primary 18E30; Secondary 18G10, 53D37
Received: 10.06.2010 and 03.12.2010

Citation: A. I. Efimov, “A proof of the Kontsevich-Soǐbel'man conjecture”, Mat. Sb., 202:4 (2011), 65–84; Sb. Math., 202:4 (2011), 527–546

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    This publication is cited in the following articles:
    1. Chan K., Ueda K., “Dual Torus Fibrations and Homological Mirror Symmetry for a(N)-Singularities”, Commun. Number Theory Phys., 7:2 (2013), 361–396  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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