Model categories of quiver representations

Henrik Holm; Peter Jørgensen

doi:10.1016/j.aim.2019.106826

Model categories of quiver representations

Henrik Holm^*, Peter Jørgensen

^*Corresponding author for this work

Department of Mathematical Sciences

3 Citations (Scopus)

6 Downloads (Pure)

Abstract

Gillespie's Theorem gives a systematic way to construct model category structures on C(M), the category of chain complexes over an abelian category M. We can view C(M) as the category of representations of the quiver ⋯→2→1→0→−1→−2→⋯ with the relations that two consecutive arrows compose to 0. This is a self-injective quiver with relations, and we generalise Gillespie's Theorem to other such quivers with relations. There is a large family of these, and following Iyama and Minamoto, their representations can be viewed as generalised chain complexes. Our result gives a systematic way to construct model category structures on many categories. This includes the category of N-periodic chain complexes, the category of N-complexes where ∂^N=0, and the category of representations of the repetitive quiver ZA_n with mesh relations.

Original language	English
Article number	106826
Journal	Advances in Mathematics
Volume	357
Number of pages	46
ISSN	0001-8708
DOIs	https://doi.org/10.1016/j.aim.2019.106826
Publication status	Published - 2019

Keywords

Abelian model categories
Chain complexes
Cotorsion pairs
Gillespie's and Hovey's Theorems
N-complexes
Periodic chain complexes

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10.1016/j.aim.2019.106826

OA-Model categoriesAccepted author manuscript, 497 KBLicence: CC BY-NC-ND

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title = "Model categories of quiver representations",

abstract = "Gillespie's Theorem gives a systematic way to construct model category structures on C(M), the category of chain complexes over an abelian category M. We can view C(M) as the category of representations of the quiver ⋯→2→1→0→−1→−2→⋯ with the relations that two consecutive arrows compose to 0. This is a self-injective quiver with relations, and we generalise Gillespie's Theorem to other such quivers with relations. There is a large family of these, and following Iyama and Minamoto, their representations can be viewed as generalised chain complexes. Our result gives a systematic way to construct model category structures on many categories. This includes the category of N-periodic chain complexes, the category of N-complexes where ∂N=0, and the category of representations of the repetitive quiver ZAn with mesh relations.",

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T1 - Model categories of quiver representations

AU - Holm, Henrik

AU - Jørgensen, Peter

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N2 - Gillespie's Theorem gives a systematic way to construct model category structures on C(M), the category of chain complexes over an abelian category M. We can view C(M) as the category of representations of the quiver ⋯→2→1→0→−1→−2→⋯ with the relations that two consecutive arrows compose to 0. This is a self-injective quiver with relations, and we generalise Gillespie's Theorem to other such quivers with relations. There is a large family of these, and following Iyama and Minamoto, their representations can be viewed as generalised chain complexes. Our result gives a systematic way to construct model category structures on many categories. This includes the category of N-periodic chain complexes, the category of N-complexes where ∂N=0, and the category of representations of the repetitive quiver ZAn with mesh relations.

AB - Gillespie's Theorem gives a systematic way to construct model category structures on C(M), the category of chain complexes over an abelian category M. We can view C(M) as the category of representations of the quiver ⋯→2→1→0→−1→−2→⋯ with the relations that two consecutive arrows compose to 0. This is a self-injective quiver with relations, and we generalise Gillespie's Theorem to other such quivers with relations. There is a large family of these, and following Iyama and Minamoto, their representations can be viewed as generalised chain complexes. Our result gives a systematic way to construct model category structures on many categories. This includes the category of N-periodic chain complexes, the category of N-complexes where ∂N=0, and the category of representations of the repetitive quiver ZAn with mesh relations.

KW - Abelian model categories

KW - Chain complexes

KW - Cotorsion pairs

KW - Gillespie's and Hovey's Theorems

KW - N-complexes

KW - Periodic chain complexes

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JO - Advances in Mathematics

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Model categories of quiver representations

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Keywords

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