A Brief Introduction to the Q-Shaped Derived Category

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A chain complex can be viewed as a representation of a certain quiver with relations, Qcpx. 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 Qcpx. It is an insight of Iyama and Minamoto that the reason D is well behaved is that, viewed as a small category, Qcpx 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, DQ. Drawing on methods of Hovey and Gillespie, we developed the theory of DQ in three recent papers. This paper offers a brief introduction to DQ, aimed at the reader already familiar with the classic derived category.

Original languageEnglish
Title of host publicationTriangulated Categories in Representation Theory and Beyond : The Abel Symposium 2022
EditorsPetter Andreas Bergh, Øyvind Solberg, Steffen Oppermann
Number of pages27
PublisherSpringer
Publication date2024
Pages141-167
ISBN (Print)9783031577888
DOIs
Publication statusPublished - 2024
EventAbel Symposium, 2022 - Ålesund, Norway
Duration: 6 Jun 202210 Jun 2022

Conference

ConferenceAbel Symposium, 2022
LandNorway
ByÅlesund
Periode06/06/202210/06/2022
SeriesAbel Symposia
Volume17
ISSN2193-2808

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

    Research areas

  • Abelian category, Abelian model category, Chain complex, Cofibration, Derived category, Exact category, Fibration, Frobenius category, Homotopy, Homotopy category, Model category, Stable category, Triangulated category, Weak equivalence

ID: 402883585