A Brief Introduction to the Q-Shaped Derived Category

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 ) 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.