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Displaying similar documents to “A note on model structures on arbitrary Frobenius categories”

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...

One-sided n -suspended categories

Jing He, Yonggang Hu, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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For an integer n 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...

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.

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...

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.

Schur-Finite Motives and Trace Identities

Alessio Del Padrone, Carlo Mazza (2009)

Bollettino dell'Unione Matematica Italiana

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We provide a sufficient condition that ensures the nilpotency of endomorphisms universally of trace zero of Schur-finite objects in a category of homological type, i.e., a -linear -category with a tensor functor to super vector spaces. We present some applications in the category of motives, where our result generalizes previous results about finite-dimensional objects, in particular by Kimura. We also present some facts which suggest that this might be the best generalization possible...

Yetter-Drinfeld-Long bimodules are modules

Daowei Lu, Shuan Hong Wang (2017)

Czechoslovak Mathematical Journal

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Let H be a finite-dimensional bialgebra. In this paper, we prove that the category ℒℛ ( H ) of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category H H * H H * 𝒴𝒟 over the tensor product bialgebra H H * as monoidal categories. Moreover if H is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.

How to construct a Hovey triple from two cotorsion pairs

James Gillespie (2015)

Fundamenta Mathematicae

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Let be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs ( , ˜ ) and ( ˜ , ) in satisfying ˜ and ˜ = ˜ . We show how to construct a (necessarily unique) abelian model structure on with (resp. ˜ ) as the class of cofibrant (resp. trivially cofibrant) objects, and (resp. ˜ ) as the class of fibrant (resp. trivially fibrant) objects.

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 Δ / G where Δ is a disjoint union of simply-laced Dynkin diagrams and G a weakly admissible group of automorphisms of Δ . Then we prove that for ‘most’ groups G , the category 𝒯 is standard, ...

Base change for Picard-Vessiot closures

Andy R. Magid (2011)

Banach Center Publications

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The differential automorphism group, over F, Π₁(F₁) of the Picard-Vessiot closure F₁ of a differential field F is a proalgebraic group over the field C F of constants of F, which is assumed to be algebraically closed of characteristic zero, and its category of C F modules is equivalent to the category of differential modules over F. We show how this group and the category equivalence behave under a differential extension E ⊃ F, where C E is also algebraically closed.

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...

Derived endo-discrete artin algebras

Raymundo Bautista (2006)

Colloquium Mathematicae

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Let Λ be an artin algebra. We prove that for each sequence ( h i ) i of non-negative integers there are only a finite number of isomorphism classes of indecomposables X b ( Λ ) , the bounded derived category of Λ, with l e n g t h E ( X ) H i ( X ) = h i for all i ∈ ℤ and E(X) the endomorphism ring of X in b ( Λ ) if and only if b ( M o d Λ ) , the bounded derived category of the category M o d Λ of all left Λ-modules, has no generic objects in the sense of [4].

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.

Colimit-dense subcategories

Jiří Adámek, Andrew D. Brooke-Taylor, Tim Campion, Leonid Positselski, Jiří Rosický (2019)

Commentationes Mathematicae Universitatis Carolinae

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Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vopěnka’s Principle, we prove that a cocomplete category is locally presentable if and only if it has a colimit dense subcategory and a generator consisting of presentable objects. We further show that a 3 -element set is colimit-dense in 𝐒𝐞𝐭 op , and spaces of countable dimension are colimit-dense in 𝐕𝐞𝐜 op .