Dg enhanced orbit categories and applications

TL;DR

This paper explores advanced constructions of cluster categories through orbit, singularity, and cosingularity frameworks, establishing universal properties and linking different categorical descriptions via Koszul duality.

Contribution

It proves the universal property of pretriangulated orbit categories and relates cluster categories as (co)singularity categories, extending prior results.

Findings

Proved the universal property of pretriangulated orbit categories.

Showed that orbit category formation commutes with dg quotients.

Connected cluster categories with (co)singularity categories via relative Koszul duality.

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.