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

Abstract

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 $\ast$, then $X$ is nothing but a colax action of the group $I$ on the dg category $X(\ast )$. 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 $X^{\prime }$, the Grothendieck construction ${\int }_I{X}^{\prime }$ is derived equivalent to ${\int }_{I}X$ if there exists a standard derived equivalence from $X^{\prime }$ 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.

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