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

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 $X^{\mathrm{\prime }}$, the Grothendieck construction ${\int }_I{X}^{\mathrm{\prime }}$ is derived equivalent to ${\int }_{I}X$ if there exists a standard derived equivalence from $X^{\mathrm{\prime }}$ 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.