A brief introduction to the Q-shaped derived category

2022/12/23 by Henrik Holm, Holm, Henrik, Peter Jørgensen +1 · 2 citations
Computer Science · Mathematics · #16E35 #18E35 #18G80 #18N40 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2212.12524

openalex publication_date 2022/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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