A Brief Introduction to the Q-Shaped Derived Category
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A Brief Introduction to the Q-Shaped Derived Category. / Holm, Henrik; Jørgensen, Peter.
Triangulated Categories in Representation Theory and Beyond: The Abel Symposium 2022. ed. / Petter Andreas Bergh; Øyvind Solberg; Steffen Oppermann. Springer, 2024. p. 141-167 (Abel Symposia, Vol. 17).Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review
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TY - GEN
T1 - A Brief Introduction to the Q-Shaped Derived Category
AU - Holm, Henrik
AU - Jørgensen, Peter
N1 - Publisher Copyright: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
KW - Abelian category
KW - Abelian model category
KW - Chain complex
KW - Cofibration
KW - Derived category
KW - Exact category
KW - Fibration
KW - Frobenius category
KW - Homotopy
KW - Homotopy category
KW - Model category
KW - Stable category
KW - Triangulated category
KW - Weak equivalence
U2 - 10.1007/978-3-031-57789-5_5
DO - 10.1007/978-3-031-57789-5_5
M3 - Article in proceedings
AN - SCOPUS:85201018245
SN - 9783031577888
T3 - Abel Symposia
SP - 141
EP - 167
BT - Triangulated Categories in Representation Theory and Beyond
A2 - Bergh, Petter Andreas
A2 - Solberg, Øyvind
A2 - Oppermann, Steffen
PB - Springer
T2 - Abel Symposium, 2022
Y2 - 6 June 2022 through 10 June 2022
ER -
ID: 402883585