Kontsevich-Soibelman猜想的证明
A proof of Kontsevich-Soibelman conjecture
AI总结：
本文在域上对本质小$A_{\infty}$-（预）范畴证明了Kontsevich-Soibelman猜想，即$A_{\infty}$-预范畴的拟等价类与具有单位态射的$A_{\infty}$-范畴的拟等价类一一对应，并构造了预三角包络且证明其拟等价不变性。
摘要：
众所周知，“Fukaya范畴”实际上是Kontsevich和Soibelman意义下的一个$A_{\infty}$-预范畴。其原因在于，一般情形下态射空间仅对横截的Lagrangian对定义，而高阶乘积仅对横截的Lagrangian序列定义。在文献中Kontsevich和Soibelman猜想：对任意分次交换环$k$，$k$上的$A_{\infty}$-预范畴的拟等价类与具有严格（或弱）单位态射的$k$上$A_{\infty}$-范畴的拟等价类之间存在一一对应。本文在$k$为域的情形下，对本质小的$A_{\infty}$-（预）范畴证明了这一猜想。特别地，由此可知我们可以将Fukaya $A_{\infty}$-预范畴替换为一个拟等价的实际$A_{\infty}$-范畴。我们还在$A_{\infty}$-预范畴框架下给出了预三角包络的自然构造，并证明了其在拟等价下的不变性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
