论文标题
前序的DG类别中的框架
Frames in Pretriangulated Dg-Categories
论文作者
论文摘要
在代数中产生的三角类别通常被描述为前二元类别的DG类别的同型类别，该类别富含具有自然的移位和锥体概念，具有与所有同源代数的所有机械相关的链综合体。 DG类别是$ \ infty $ - 类别的代数模型，因此适合于更高类别模型的广泛生态系统及其之间的翻译。在本文中，我们描述了两种方法之间的等效性，以将前研究的DG类别转换为QuasicateGory。 DG类别的DG-temnve是一种准术，其简形是映射复合物中地图的连贯家族。相比之下，前序类别的循环类别忘记了映射络合物的所有高度元素，但成为间接编码同位素结构的联合启动类别。然后，该联结类别具有相关的框架准游戏，其中简单是芦苇构成的分辨率。 对于前培养的DG类别的DG-never中的每个单纯形，我们构建了这种乳纤维成型的分辨率，然后证明该构建定义了类准游戏的等效性，这是自然而然的简单同型。我们的构造足以进行计算，并提供了对帧帧构图的分辨率的直观解释，作为映射缸的概括。
Triangulated categories arising in algebra can often be described as the homotopy category of a pretriangulated dg-category, a category enriched in chain complexes with a natural notion of shifts and cones that is accessible with all the machinery of homological algebra. Dg-categories are algebraic models of $\infty$-categories and thus fit into a wide ecosystem of higher-categorical models and translations between them. In this paper we describe an equivalence between two methods to turn a pretriangulated dg-category into a quasicategory. The dg-nerve of a dg-category is a quasicategory whose simplices are coherent families of maps in the mapping complexes. In contrast, the cycle category of a pretriangulated category forgets all higher-degree elements of the mapping complexes but becomes a cofibration category that encodes the homotopical structure indirectly. This cofibration category then has an associated quasicategory of frames in which simplices are Reedy-cofibrant resolutions. For every simplex in the dg-nerve of a pretriangulated dg-category we construct such a Reedy-cofibrant resolution and then prove that this construction defines an equivalence of quasicategories which is natural up to simplicial homotopy. Our construction is explicit enough for calculations and provides an intuitive explanation of the resolutions in the quasicategory of frames as a generalisation of the mapping cylinder.
