On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions

Hans-Jürgen Vogel

Displaying similar documents to “On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions”

Schur-Finite Motives and Trace Identities

Alessio Del Padrone, Carlo Mazza (2009)

Bollettino dell'Unione Matematica Italiana

Similarity:

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 $\mathbb{Q}$ -linear $\otimes$ -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...

$n$ -angulated quotient categories induced by mutation pairs

Zengqiang Lin (2015)

Czechoslovak Mathematical Journal

Similarity:

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...

The categories of presheaves containing any category of algebras

V. Trnková, J. Reiterman

Similarity:

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

One-sided $n$ -suspended categories

Jing He, Yonggang Hu, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

Similarity:

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...

Gorenstein dimension of abelian categories arising from cluster tilting subcategories

Yu Liu, Panyue Zhou (2021)

Czechoslovak Mathematical Journal

Similarity:

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...

A note on model structures on arbitrary Frobenius categories

Zhi-wei Li (2017)

Czechoslovak Mathematical Journal

Similarity:

We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category $ℱ$ such that the homotopy category of this model structure is equivalent to the stable category $\underset{̲}{ℱ}$ as triangulated categories. This seems to be well-accepted by experts but we were unable to find a complete proof for it in the literature. When $ℱ$ is a weakly idempotent complete (i.e., every split monomorphism is an inflation) Frobenius category, the model structure we constructed is an exact...

Recollement of colimit categories and its applications

Ju Huang, QingHua Chen, Chunhuan Lai (2020)

Czechoslovak Mathematical Journal

Similarity:

We give an explicit recollement for a cocomplete abelian category and its colimit category. We obtain some applications on Leavitt path algebras, derived equivalences and $K$ -groups.

Colimit-dense subcategories

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

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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 ${𝐕𝐞𝐜}^{op}$ .

Yetter-Drinfeld-Long bimodules are modules

Daowei Lu, Shuan Hong Wang (2017)

Czechoslovak Mathematical Journal

Similarity:

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 $_{H \otimes H^{*}}^{H \otimes H^{*}} 𝒴𝒟$ over the tensor product bialgebra $H \otimes H^{*}$ 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.

Two results of $n$ -exangulated categories

Jian He, Jing He, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

Similarity:

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.

Base change for Picard-Vessiot closures

Andy R. Magid (2011)

Banach Center Publications

Similarity:

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.

Various categorical approaches to statistical spaces

Tadeusz Bromek, Maria Moszyńska

Similarity:

CONTENTSIntroduction..............................................................................................................51. Preliminaries........................................................................................................61.0. Measurable, probabilistic, and statistical spaces..............................................61.1. Transition functions..........................................................................................61.2. Linear space of...

Triangulated categories of periodic complexes and orbit categories

Jian Liu (2023)

Czechoslovak Mathematical Journal

Similarity:

We investigate the triangulated hull of orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull corresponds to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao’s result on the finite dimensional algebra of finite global dimension. As the first application, if $A$ , $B$ are flat algebras...

On $n$ -exact categories

Said Manjra (2019)

Czechoslovak Mathematical Journal

Similarity:

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 uniquely partitionable relational structures and object systems

Jozef Bucko, Peter Mihók (2006)

Discussiones Mathematicae Graph Theory

Similarity:

We introduce object systems as a common generalization of graphs, hypergraphs, digraphs and relational structures. Let C be a concrete category, a simple object system over C is an ordered pair S = (V,E), where E = A₁,A₂,...,Aₘ is a finite set of the objects of C, such that the ground-set $V (A_{i})$ of each object $A_{i} \in E$ is a finite set with at least two elements and $V \supseteq ⋃_{i = 1}^{m} V (A_{i})$ . To generalize the results on graph colourings to simple object systems we define, analogously as for graphs, that an additive induced-hereditary...