论文标题
DG类别图及其粘合的标准派生等效性的表征
Characterizations of standard derived equivalences of diagrams of dg categories and their gluings
论文作者
论文摘要
在本文中，由小型类别$ i $ $ x $提出的差分分级（DG）和DG函子的图表是由小型DG类别的2类K-DGCAT，DG Foundors和DG自然转换的2类k-DGCAT，用于固定通勤环$ \ MATHBB {K {K {K} $。如果$ i $是一个只有一个对象$*$的类别的组，则$ x $不过是DG类别$ x（*）$的组$ i $的Colax诉讼。从这个意义上讲，这个$ x $可以被视为具有小组的Colax行动的DG类别的概括。我们通过通过组动作概括了DG类别之间的相应概念来定义此类colax函子之间标准得出的等效性的概念。我们的第一个主要结果给出了此概念的一些特征，其中一个是根据倾斜对象的广义版本和准等效性给出的。另一方面，对于这样的Colax Foundator $ X $，DG类别$ x（i）$带有$ i $ $ i $的$ i $对象可以粘合在一起，以拥有单个DG类别$ \ int x $，称为$ x $的grothendieck构造。我们的第二个主要结果坚持认为，对于此类Colax Foundators $ X $和$ X'$，Grothendieck Construction $ \ int X'$的派生等于$ \ int x $，如果存在标准派生在$ x'$到$ x $的标准等价。这些结果分别将\ cite {asa-a}和\ cite {asa-13}的主要结果概括为DG情况。这些都是新的，即使对于具有小组动作的DG类别。特别是，第二个结果给出了一个新工具，可以显示DG类别的轨道类别与小组动作之间的派生等效性，这将在某些示例中进行说明。
A diagram consisting of differential graded (dg for short) categories and dg functors is formulated in this paper as a colax functor $X$ from a small category $I$ to the 2-category k-dgCat of small dg categories, dg functors and dg natural transformations for a fixed commutative ring $\mathbb{k}$. If $I$ is a group regarded as a category with only one object $*$, then $X$ is nothing but a colax action of the group $I$ on the dg category $X(*)$. In this sense, this $X$ can be regarded as a generalization of a dg category with a colax action of a group. We define a notion of standard derived equivalence between such colax functors by generalizing the corresponding notion between dg categories with a group action. Our first main result gives some characterizations of this notion, one of which is given in terms of generalized versions of a tilting object and a quasi-equivalence. On the other hand, for such a colax functor $X$, the dg categories $X(i)$ with $i$ objects of $I$ can be glued together to have a single dg category $\int X$, called the Grothendieck construction of $X$. Our second main result insists that for such colax functors $X$ and $X'$, the Grothendieck construction $\int X'$ is derived equivalent to $\int X$ if there exists a standard derived equivalence from $X'$ to $X$. These results generalize the main results of \cite{Asa-a} and \cite{Asa-13} to the dg case, respectively. These are new even for dg categories with group actions. In particular, the second result gives a new tool to show the derived equivalence between the orbit categories of dg categories with group actions, which will be illustrated in some examples.
