arXiv 2201.107602022-01math.RTmath.CTmath.RA

图示dg范畴的标准导出等价性的特征及其粘合

Characterizations of standard derived equivalences of diagrams of dg categories and their gluings

Shizuoka University（静冈大学）；Kyoto University（京都大学）；Osaka Central Advanced Mathematical Institute（大阪中央先进数学研究所）；Beijing Jiaotong University（北京交通大学）

Hideto Asashiba, Shengyong Pan

备注：: Fix some errors,more details, colimit changes to co-Yoneda,all the comments are welcome

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AI总结：

本文研究了dg范畴图示的colax作用下的标准导出等价性及其粘合构造，提出了新的工具用于证明轨道范畴之间的导出等价性。

摘要：

本文中，将由微分格（dg）范畴和dg函子组成的图示定义为从一个小范畴I到k-dgCat 2-范畴的colax函子X，其中k-dgCat是小dg范畴、dg函子和dg自然变换的2-范畴，其定义在固定的交换环k上。如果I被视为只有单个对象*的范畴，则X其实就是群I对dg范畴X(*)的colax作用。因此，这种X可以被视为群的colax作用下的dg范畴的推广。我们通过推广dg范畴与群作用之间的相应概念，定义了此类colax函子之间的标准导出等价性。我们的第一个主要结果给出了一些这种概念的特征，其中之一是通过广义的倾斜对象和准等价的广义版本来描述。另一方面，对于这样的colax函子X，dg范畴X(i)（i是I的对象）可以粘合在一起，形成一个单一的dg范畴∫_I X，称为X的Grothendieck构造。我们的第二个主要结果断言，对于这样的colax函子X和X'，如果存在从X'到X的标准导出等价性，则Grothendieck构造∫_I X'与∫_I X导出等价。这些结果分别将第一作者的结果推广到dg情况。即使对于具有群作用的dg范畴，这些结果也是新的。特别是，第二个结果提供了一种新的工具，用于证明具有群作用的dg范畴的轨道范畴之间的导出等价性，这将在一些例子中得到说明。

英文摘要：

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 $\mathbf{k}$-dgCat of small dg categories, dg functors and dg natural transformations over a fixed commutative ring $\mathbf{k}$. 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 $\int_I X$, called the Grothendieck construction of $X$. Our second main result asserts that for such colax functors $X$ and $X'$, the Grothendieck construction $\int_I X'$ is derived equivalent to $\int_I X$ if there exists a standard derived equivalence from $X'$ to $X$. These results generalize the first-named author's results to the dg case, respectively. Even for dg categories with group actions, these results are new. 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.

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