A proof of Kontsevich-Soibelman conjecture
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2009
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| _version_ | 1866912221419798528 |
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| author | Efimov, Alexander I. |
| author_facet | Efimov, Alexander I. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0911_0123 |
| institution | arXiv |
| publishDate | 2009 |
| record_format | arxiv |
| spellingShingle | A proof of Kontsevich-Soibelman conjecture Efimov, Alexander I. Category Theory Symplectic Geometry 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. |
| title | A proof of Kontsevich-Soibelman conjecture |
| topic | Category Theory Symplectic Geometry |
| url | https://arxiv.org/abs/0911.0123 |