On the structure of triangulated categories with finitely many indecomposables

Claire Amiot

Displaying similar documents to “On the structure of triangulated categories with finitely many indecomposables”

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.

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...

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...

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§...

Stable tubes in extriangulated categories

Li Wang, Jiaqun Wei, Haicheng Zhang (2022)

Czechoslovak Mathematical Journal

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Let $𝒳$ be a semibrick in an extriangulated category. If $𝒳$ is a $τ$ -semibrick, then the Auslander-Reiten quiver $Γ (ℱ (𝒳))$ of the filtration subcategory $ℱ (𝒳)$ generated by $𝒳$ is $ℤ 𝔸_{\infty}$ . If $𝒳 = {X_{i}}_{i = 1}^{t}$ is a $τ$ -cycle semibrick, then $Γ (ℱ (𝒳))$ is $ℤ 𝔸_{\infty} / τ_{𝔸}^{t}$ .

Two results of $n$ -exangulated categories

Jian He, Jing He, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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M. Herschend, Y. Liu, H. Nakaoka introduced $n$ -exangulated categories, which are a simultaneous generalization of $n$ -exact categories and $(n + 2)$ -angulated categories. This paper consists of two results on $n$ -exangulated categories: (1) we give an equivalent characterization of axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an $n$ -exangulated category.

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...

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...

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.

On almost complex structures from classical linear connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $ℳ f_{m}$ be the category of $m$ -dimensional manifolds and local diffeomorphisms and let $T$ be the tangent functor on $ℳ f_{m}$ . Let $𝒱$ be the category of real vector spaces and linear maps and let $𝒱_{m}$ be the category of $m$ -dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F : 𝒱_{m} \to 𝒱$ admitting $ℳ f_{m}$ -natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$ -dimensional manifolds $M$ into almost complex structures $\tilde{J} (\nabla)$ on $F (T) M = ⋃_{x \in M} F (T_{x} M)$ .

The Roquette category of finite $p$ -groups

Serge Bouc (2015)

Journal of the European Mathematical Society

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Let $p$ be a prime number. This paper introduces the Roquette category $ℛ_{p}$ of finite $p$ -groups, which is an additive tensor category containing all finite $p$ -groups among its objects. In $ℛ_{p}$ , every finite $p$ -group $P$ admits a canonical direct summand $\partial P$ , called the edge of $P$ . Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$ -groups, and the tensor structure of $ℛ_{p}$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors...

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...

Quasicontinuous spaces

Jing Lu, Bin Zhao, Kaiyun Wang, Dong Sheng Zhao (2022)

Commentationes Mathematicae Universitatis Carolinae

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We lift the notion of quasicontinuous posets to the topology context, called quasicontinuous spaces, and further study such spaces. The main results are: (1) A $T_{0}$ space $(X, τ)$ is a quasicontinuous space if and only if $S I (X)$ is locally hypercompact if and only if $(τ_{S I}, \subseteq)$ is a hypercontinuous lattice; (2) a $T_{0}$ space $X$ is an $S I$ -continuous space if and only if $X$ is a meet continuous and quasicontinuous space; (3) if a $C$ -space $X$ is a well-filtered poset under its specialization order, then $X$ is a quasicontinuous...

$T_{2}$ 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...

$n$ -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...

Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiří Rosický (2015)

Journal of the European Mathematical Society

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We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $𝒮$ of morphisms in a locally presentable category $𝒞$ of structures, the orthogonal class of objects...