关于三角剖分模型的变形
On deformations of triangulated models
摘要：
本文是一个项目的第一部分，旨在了解变形的三角范畴，更准确地说，他们的dg和A无穷大模型，并将由此产生的理论应用到模型中发生的同调镜像对称设置。在这第一篇论文中，我们专注于模型的派生和相关的类别，基于扭曲的对象的dg或$A_{\infty}$ -代数的经典建设。对于这样一个模型的Hochschild 2上循环，我们描述了相应的“曲率补偿”变形，可以完全理解的扭曲对象的框架内。我们解开的具体情况下，派生的A无穷大和阿贝尔范畴，同伦范畴，和类别的分次自由qdg-模块的建设。我们在模型上确定了一个纯度条件，以确保模型的结构在变形下保持不变。这个条件通常对于同伦范畴是满足的，但对于无界的导出范畴则不满足。
英文摘要：
This paper is the first part of a project aimed at understanding deformations of triangulated categories, and more precisely their dg and A infinity models, and applying the resulting theory to the models occurring in the Homological Mirror Symmetry setup. In this first paper, we focus on models of derived and related categories, based upon the classical construction of twisted objects over a dg or $A_{\infty}$-algebra. For a Hochschild 2 cocycle on such a model, we describe a corresponding "curvature compensating" deformation which can be entirely understood within the framework of twisted objects. We unravel the construction in the specific cases of derived A infinity and abelian categories, homotopy categories, and categories of graded free qdg-modules. We identify a purity condition on our models which ensures that the structure of the model is preserved under deformation. This condition is typically fulfilled for homotopy categories, but not for unbounded derived categories.
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