A brief introduction to the $Q$-shaped derived category

TL;DR

This paper introduces the $Q$-shaped derived category, a generalization of the classic derived category viewed through the lens of quivers with relations possessing a Serre functor, expanding the theoretical framework.

Contribution

It provides a concise introduction to the $Q$-shaped derived category, extending the classic derived category concept to new quivers with Serre functors, based on recent theoretical developments.

Findings

Generalization of derived categories to $Q$-shaped versions.

Connection between Serre functors and well-behaved categories.

Foundation for further research in $Q$-shaped derived categories.

Abstract

A chain complex can be viewed as a representation of a certain quiver with relations, $Q^{\operatorname{cpx}}$. The vertices are the integers, there is an arrow $q \xrightarrow{} q-1$ for each integer $q$, and the relations are that consecutive arrows compose to $0$. Hence the classic derived category $\mathscr{D}$ can be viewed as a category of representations of $Q^{\operatorname{cpx}}$. It is an insight of Iyama and Minamoto that the reason $\mathscr{D}$ is well behaved is that, viewed as a small category, $Q^{\operatorname{cpx}}$ has a Serre functor. Generalising the construction of $\mathscr{D}$ to other quivers with relations which have a Serre functor results in the $Q$-shaped derived category ${\mathscr{D}}_Q$. Drawing on methods of Hovey and Gillespie, we developed the theory of ${\mathscr{D}}_Q$ in three recent papers. This paper offers a brief introduction to ${\mathscr{D}}_Q$, aimed at the reader already familiar with the classic derived category.