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