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- W4226402270 abstract "A diagram consisting of differential graded (dg for short) categories and dg functors is formulated as a colax functor $X$ from a small category $I$ to the 2-category $kdgCat$ of small dg categories, dg functors and dg natural transformations for a fixed commutative ring $k$. 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 without an assumption of $k$-flatness (or $k$-projectivity) on $X$, 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 $Gr(X)$, called the Grothendieck construction of $X$. % Our second main result insists that for such colax functors $X$ and $X'$, the Grothendieck construction $Gr(X')$ is derived equivalent to $Gr(X)$ 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 a group action. In particular, the second result gives a new tool to show the derived equivalence between the orbit categories of dg categories with a group action, which will be illustrated in some examples." @default.
- W4226402270 created "2022-05-05" @default.
- W4226402270 creator A5061812330 @default.
- W4226402270 creator A5087040739 @default.
- W4226402270 date "2022-01-26" @default.
- W4226402270 modified "2023-10-18" @default.
- W4226402270 title "Characterizations of standard derived equivalences of diagrams of dg categories and their gluings" @default.
- W4226402270 doi "https://doi.org/10.48550/arxiv.2201.10760" @default.
- W4226402270 hasPublicationYear "2022" @default.
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