关于A-无穷代数、A-无穷范畴与非交换几何的札记 I
Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I
AI总结：
本文基于分次向量空间范畴的形式概形概念，提出研究A-infinity代数与范畴的几何方法，厘清相关结构问题，推广多个经典猜想与概念，并给出对应二维拓扑场论的结构等价性结论。
摘要：
我们基于分次向量空间范畴中形式概形的概念，提出了研究A-infinity代数与A-infinity范畴的几何方法。该几何方法厘清了若干问题，如同调单位的概念，或是A-infinity函子上的A-infinity结构。我们从几何视角讨论了A-infinity代数的Hochschild复形。本文还包含射影簇的正常性与光滑性的同调版本，以及Hodge到de Rham退化猜想的非交换版本。我们也讨论了Deligne猜想的一种推广，该推广同时涵盖Hochschild链与上链。在论文末尾，我们描述了二维曲面拓扑PROP的奇异链所构成的PROP，在带内积的A-infinity代数的Hochschild链复形上的作用。该作用本质上等价于与Calabi-Yau范畴相关的二维拓扑场论结构。
英文摘要：
We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne's conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product. This action is essentially equivalent to the structure of 2-dimensional Topological Field Theory associated with a Calabi-Yau category.
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