A Brief Introduction to the Q-Shaped Derived Category

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

Standard

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

Harvard

Holm, H & Jørgensen, P 2024, A Brief Introduction to the Q-Shaped Derived Category. in PA Bergh, Ø Solberg & S Oppermann (eds), Triangulated Categories in Representation Theory and Beyond: The Abel Symposium 2022. Springer, Abel Symposia, vol. 17, pp. 141-167, Abel Symposium, 2022, Ålesund, Norway, 06/06/2022. https://doi.org/10.1007/978-3-031-57789-5_5

APA

Holm, H., & Jørgensen, P. (2024). A Brief Introduction to the Q-Shaped Derived Category. In P. A. Bergh, Ø. Solberg, & S. Oppermann (Eds.), Triangulated Categories in Representation Theory and Beyond: The Abel Symposium 2022 (pp. 141-167). Springer. Abel Symposia Vol. 17 https://doi.org/10.1007/978-3-031-57789-5_5

Vancouver

Holm H, Jørgensen P. A Brief Introduction to the Q-Shaped Derived Category. In Bergh PA, Solberg Ø, Oppermann S, editors, Triangulated Categories in Representation Theory and Beyond: The Abel Symposium 2022. Springer. 2024. p. 141-167. (Abel Symposia, Vol. 17). https://doi.org/10.1007/978-3-031-57789-5_5

Author

Holm, Henrik ; Jørgensen, Peter. / A Brief Introduction to the Q-Shaped Derived Category. Triangulated Categories in Representation Theory and Beyond: The Abel Symposium 2022. editor / Petter Andreas Bergh ; Øyvind Solberg ; Steffen Oppermann. Springer, 2024. pp. 141-167 (Abel Symposia, Vol. 17).

Bibtex

@inproceedings{7ad93d76627c46f5bf715eec1d1e5df8,
title = "A Brief Introduction to the Q-Shaped Derived Category",
abstract = "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.",
keywords = "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",
author = "Henrik Holm and Peter J{\o}rgensen",
note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.; Abel Symposium, 2022 ; Conference date: 06-06-2022 Through 10-06-2022",
year = "2024",
doi = "10.1007/978-3-031-57789-5_5",
language = "English",
isbn = "9783031577888",
series = "Abel Symposia",
pages = "141--167",
editor = "Bergh, {Petter Andreas} and {\O}yvind Solberg and Steffen Oppermann",
booktitle = "Triangulated Categories in Representation Theory and Beyond",
publisher = "Springer",
address = "Switzerland",

}

RIS

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