A proof of the Kontsevich-So\v{\i}bel'man conjecture

A. I. Efimov

Sbornik. Mathematics, Tome 202 (2011) no. 4, pp. 527-546

Résumé

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.

Détail

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