Two results of $n$-exangulated categories

Jian He; Jing He; Panyue Zhou

Displaying similar documents to “Two results of $n$ -exangulated categories”

One-sided $n$ -suspended categories

Jing He, Yonggang Hu, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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For an integer $n \geq 3$ , we introduce a simultaneous generalization of $(n - 2)$ -exact categories and $n$ -angulated categories, referred to as one-sided $n$ -suspended categories. Notably, one-sided $n$ -angulated categories are specific instances of this structure. We establish a framework for transitioning from these generalized categories to their $n$ -angulated counterparts. Additionally, we present a method for constructing $n$ -angulated quotient categories from Frobenius $n$ -prile categories. Our results unify...

$n$ -angulated quotient categories induced by mutation pairs

Zengqiang Lin (2015)

Czechoslovak Mathematical Journal

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Geiss, Keller and Oppermann (2013) introduced the notion of $n$ -angulated category, which is a “higher dimensional” analogue of triangulated category, and showed that certain $(n - 2)$ -cluster tilting subcategories of triangulated categories give rise to $n$ -angulated categories. We define mutation pairs in $n$ -angulated categories and prove that given such a mutation pair, the corresponding quotient category carries a natural $n$ -angulated structure. This result generalizes a theorem of Iyama-Yoshino...

On the structure of triangulated categories with finitely many indecomposables

Claire Amiot (2007)

Bulletin de la Société Mathématique de France

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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$ . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $𝒯$ is of the form $ℤ Δ / G$ where $Δ$ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $ℤ Δ$ . Then we prove that for ‘most’ groups $G$ , the category $𝒯$ is standard, ...

Limits and colimits in certain categories of spaces of continuous functions

Marvin W. Grossman

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CONTENTSIntroduction................................................................................................................................................................................5§ 1. Notation and preliminaries.............................................................................................................................................6§ 2. Epimorphisms and monomorphisms.........................................................................................................................7§...

On $n$ -exact categories

Said Manjra (2019)

Czechoslovak Mathematical Journal

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An $n$ -exact category is a pair consisting of an additive category and a class of sequences with $n + 2$ terms satisfying certain axioms. We introduce $n$ -weakly idempotent complete categories. Then we prove that an additive $n$ -weakly idempotent complete category together with the class $𝒞_{n}$ of all contractible sequences with $n + 2$ terms is an $n$ -exact category. Some properties of the class $𝒞_{n}$ are also discussed.

Higher-dimensional Auslander-Reiten sequences

Jiangsha Li, Jing He (2024)

Czechoslovak Mathematical Journal

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Zhou and Zhu have shown that if $𝒞$ is an $(n + 2)$ -angulated category and $𝒳$ is a cluster tilting subcategory of $𝒞$ , then the quotient category $𝒞 / 𝒳$ is an $n$ -abelian category. We show that if $𝒞$ has Auslander-Reiten $(n + 2)$ -angles, then $𝒞 / 𝒳$ has Auslander-Reiten $n$ -exact sequences.

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer $n$ , let $A_{n}$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $ℳ_{n + 1}$ and $ℱ_{n}$ , where $ℳ_{n + 1}$ and $ℱ_{n}$ are the two extriangulated categories corresponding to the representation category of $A_{n + 1}$ and the morphism category of projective representations of $A_{n}$ , respectively. As a...

On category $𝒪$ for cyclotomic rational Cherednik algebras

Iain G. Gordon, Ivan Losev (2014)

Journal of the European Mathematical Society

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We study equivalences for category $𝒪_{p}$ of the rational Cherednik algebras $𝐇_{p}$ of type $G_{ℓ} (n) = {(μ_{ℓ})}^{n} ⋊ 𝔖_{n}$ : a highest weight equivalence between $𝒪_{p}$ and $𝒪_{σ (p)}$ for $σ \in 𝔖_{ℓ}$ and an action of $𝔖_{ℓ}$ on an explicit non-empty Zariski open set of parameters $p$ ; a derived equivalence between $𝒪_{p}$ and $𝒪_{p^{'}}$ whenever $p$ and $p^{'}$ have integral difference; a highest weight equivalence between $𝒪_{p}$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$ . As a consequence, we confirm special cases...

Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Justyna Kosakowska, Daniel Simson (2008)

Colloquium Mathematicae

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Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor $ℍ_{C} : C - C o m o d \to H_{C} - C o m o d$ that restricts to a representation equivalence $ℍ_{C} : C - c o m o d \to H_{C} - c o m o d_{s p}^{•}$ , where $H_{C}$ is a coradical square complete hereditary...

$T_{2}$ and $T_{3}$ objects at $p$ in the category of proximity spaces

Muammer Kula, Samed Özkan (2020)

Mathematica Bohemica

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In previous papers, various notions of pre-Hausdorff, Hausdorff and regular objects at a point $p$ in a topological category were introduced and compared. The main objective of this paper is to characterize each of these notions of pre-Hausdorff, Hausdorff and regular objects locally in the category of proximity spaces. Furthermore, the relationships that arise among the various $Pre T_{2}$ , $T_{i}$ , $i = 0, 1, 2, 3$ , structures at a point $p$ are investigated. Finally, we examine the relationships between the generalized...

The categories of presheaves containing any category of algebras

V. Trnková, J. Reiterman

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ContentsIntroduction.................................................................................................................................................. 5I. Preliminaries........................................................................................................................................... 6II. Main theorem.......................................................................................................................................... 8III. The...

Gorenstein dimension of abelian categories arising from cluster tilting subcategories

Yu Liu, Panyue Zhou (2021)

Czechoslovak Mathematical Journal

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Let $𝒞$ be a triangulated category and $𝒳$ be a cluster tilting subcategory of $𝒞$ . Koenig and Zhu showed that the quotient category $𝒞 / 𝒳$ is Gorenstein of Gorenstein dimension at most one. But this is not always true when $𝒞$ becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let $𝒞$ be an extriangulated category with enough projectives and enough injectives, and...

Cotorsion pairs in comma categories

Yuan Yuan, Jian He, Dejun Wu (2024)

Czechoslovak Mathematical Journal

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Let $𝒜$ and $ℬ$ be abelian categories with enough projective and injective objects, and $T : 𝒜 \to ℬ$ a left exact additive functor. Then one has a comma category $(ℬ ↓ T)$ . It is shown that if $T : 𝒜 \to ℬ$ is $𝒳$ -exact, then $(^{⊥} 𝒳, 𝒳)$ is a (hereditary) cotorsion pair in $𝒜$ and $(^{⊥} 𝒴, 𝒴)$ ) is a (hereditary) cotorsion pair in $ℬ$ if and only if $((\binom{^{⊥} 𝒳}{^{⊥} 𝒴})), 〈 𝐡 (𝒳, 𝒴) 〉)$ is a (hereditary) cotorsion pair in $(ℬ ↓ T)$ and $𝒳$ and $𝒴$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $𝒜$ and $ℬ$ can induce special preenveloping...

Displaying similar documents to “Two results of n -exangulated categories”

Displaying similar documents to “Two results of $n$ -exangulated categories”