n-Exangulated categories (II): Constructions from n-cluster tilting subcategories

Herschend, Martin; Liu, Yu; Nakaoka, Hiroyuki

doi:10.1016/j.jalgebra.2021.11.042

Cited by 19 publications

(16 citation statements)

References 20 publications

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“…(2) From [4, Proposition 4.34] and [4, Proposition 4.5], we know that (n + 2)-angulated in the sense of Geiss-Keller-Oppermann [3] and n-exact categories in the sense of Jasso [7] are n-exangulated categories. There are some other examples of n-exangulated categories which are neither n-exact nor (n + 2)-angulated, see [4][5][6]8].…”

Section: Preliminariesmentioning

confidence: 99%

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Localization of n-exangulated categories

He¹,

He²,

Zhou³

2022

Preprint

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Nakaoka-Ogawa-Sakai considered the localization of an extriangulated category. This construction unified the Serre quotient of abelian categories and the Verdier quotient of triangulated categories. Recently, Herschend-Liu-Nakaoka defined n-exangulated categories as a higher dimensional analogue of extriangulated categories. Let C be an n-exangulated category and F be a multiplicative system satisfying mild assumption. In this article, we give a necessary and sufficient condition for the localization of C be an n-exangulated category. This way gives a new class of n-exangulated categories which are neither n-exact nor (n + 2)-angulated in general. Moreover, our result also generalizes work by Nakaoka-Ogawa-Sakai.

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Section: Preliminariesmentioning

confidence: 99%

“…As typical examples we know that n-exact categories and (n + 2)-angulated categories are n-exangulated categories, see [4,Proposition 4.34] and [4,Proposition 4.5]. However, there are some other examples of n-exangulated categories which are neither n-exact nor (n + 2)-angulated, see [4][5][6]8].…”

Section: Introductionmentioning

confidence: 99%

Localization of n-exangulated categories

He¹,

He²,

Zhou³

2022

Preprint

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“…2] in briefly recalling the definition of an n-exangulated category, which is a higher analogue of an extriangulated category as introduced in [54]. See also [27].…”

Section: Preliminaries On N-exangulated Categoriesmentioning

confidence: 99%

“…3.8] and its dual, we have that both A-inflations and A-deflations are closed under composition. This implies that (A, E A , s A ) is an n-exangulated category by [27,Prop. 4.2(2)].…”

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confidence: 94%

The category of extensions and a characterisation of $n$-exangulated functors

Bennett-Tennenhaus¹,

Haugland²,

Sandøy³

et al. 2022

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Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information. We study how functors between categories of extensions relate to those at the level of the original categories. When the additive categories in question are n-exangulated, this leads to a characterisation of n-exangulated functors.Our approach enables us to study n-exangulated categories from a 2-categorical perspective. We introduce n-exangulated natural transformations and characterise them using categories of extensions. Our characterisations allow us to establish a 2-functor between the 2-categories of small n-exangulated categories and small exact categories. A similar result with no smallness assumption is also proved.We employ our theory to produce various examples of n-exangulated functors and natural transformations. Although the motivation for this article stems from representation theory and the study of n-exangulated categories, our results are widely applicable: several require only an additive category equipped with a biadditive functor with no extra assumptions; others can be applied by endowing an additive category with its split n-exangulated structure.

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“…As typical examples we know that n-exact categories and (n + 2)-angulated categories are n-exangulated categories, see [1,Proposition 4.34] and [1,Proposition 4.5]. However, there are some other examples of n-exangulated categories which are neither n-exact nor (n + 2)-angulated, see [1][2][3]6].…”

Section: Introductionmentioning

confidence: 99%