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 −→ q − 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.

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