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.
Categories: math.RT math.CT