Dg enhanced orbit categories and applications
Li Fan, Bernhard Keller, Yu Qiu
TL;DR
The paper develops a comprehensive dg/theory of orbit categories, proving a universal property for pretriangulated dg orbit categories and showing that orbit formation commutes with dg quotients; these results underpin applications to higher cluster categories arising from differential bigraded Calabi--Yau completions. It then connects cluster categories with (co)singularity categories by constructing enlarged cluster categories and shrunk singularity categories and relating them via a relative Koszul duality framework. The work generalizes the construction of $m$-cluster categories beyond hereditary settings and provides a robust link between dg orbit categories, Verdier quotients, and bigraded structures, including a detailed treatment of the $N$-reduction and degree-collapse phenomena. The findings yield new perspectives on how perfect derived categories can be realized as orbit- or cosingularity- constructions and establish exact compatibilities that facilitate translating between CY completions, relative dualities, and cluster- and singularity- theoretic data. Overall, the paper broadens the scope of cluster theory to dg- and dbg-enhanced contexts, with concrete implications for higher Calabi–Yau structures and their categorical realizations.
Abstract
Our aim in this paper is to prove two results related to the three constructions of cluster categories: as orbit categories, as singularity categories and as cosingularity categories. In the first part of the paper, we prove the universal property of pretriangulated orbit categories of dg categories first stated by the second-named author in 2005. We deduce that the passage to an orbit category commutes with suitable dg quotients. We apply these results to study collapsing of grading for (higher) cluster categories constructed from bigraded Calabi-Yau completions as introduced by Ikeda-Qiu. The second part of the paper is motivated by the construction of cluster categories as (co)singularity categories. We show that, for any dg algebra $A$, its perfect derived category can be realized in two ways: firstly, as an (enlarged) cluster category of a certain differential bigraded algebra, generalizing a result of Ikeda-Qiu, and secondly as a (shrunk) singularity category of another differential bigraded algebra, generalizing a result of Happel following Hanihara. We relate these two descriptions using a version of relative Koszul duality.