Derived functors of $\varprojlim $ and abelian ab3*- and Ab4*-categories with enough injectives

Piotr Ossowski

Displaying similar documents to “Derived functors of $lim_{\leftarrow}$ and abelian ab3- and Ab4-categories with enough injectives”

Recollement of colimit categories and its applications

Ju Huang, QingHua Chen, Chunhuan Lai (2020)

Czechoslovak Mathematical Journal

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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.

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 $\mathbb{Q}$ -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

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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...

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...

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 $(, \tilde{})$ and $(\tilde{},)$ in satisfying $\tilde{} \subseteq$ and $\cap \tilde{} = \tilde{} \cap$ . We show how to construct a (necessarily unique) abelian model structure on with (resp. $\tilde{}$ ) as the class of cofibrant (resp. trivially cofibrant) objects, and (resp. $\tilde{}$ ) as the class of fibrant (resp. trivially fibrant) objects.

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...

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...

A note on model structures on arbitrary Frobenius categories

Zhi-wei Li (2017)

Czechoslovak Mathematical Journal

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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...

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

Hans-Jürgen Vogel (2001)

Discussiones Mathematicae - General Algebra and Applications

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The category of all binary relations between arbitrary sets turns out to be a certain symmetric monoidal category Rel with an additional structure characterized by a family $d = (d_{A} : A \to A ⨂ A | A \in | R e l |)$ of diagonal morphisms, a family $t = (t_{A} : A \to I | A \in | R e l |)$ of terminal morphisms, and a family $\nabla = (\nabla_{A} : A ⨂ A \to A | A \in | R e l |)$ of diagonal inversions having certain properties. Using this properties in [11] was given a system of axioms which characterizes the abstract concept of a halfdiagonal-halfterminal-symmetric monoidal category with diagonal inversions (hdht∇s-category)....

Relative Auslander bijection in $n$ -exangulated categories

Jian He, Jing He, Panyue Zhou (2023)

Czechoslovak Mathematical Journal

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The aim of this article is to study the relative Auslander bijection in $n$ -exangulated categories. More precisely, we introduce the notion of generalized Auslander-Reiten-Serre duality and exploit a bijection triangle, which involves the generalized Auslander-Reiten-Serre duality and the restricted Auslander bijection relative to the subfunctor. As an application, this result generalizes the work by Zhao in extriangulated categories.

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.

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 $_{H \otimes H^{*}}^{H \otimes H^{*}} 𝒴𝒟$ 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.

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 ${𝐕𝐞𝐜}^{op}$ .

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.

On a generalization of Abelian sequential groups

Saak S. Gabriyelyan (2013)

Fundamenta Mathematicae

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Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G, τ)}^{\land}$ is a dense -closed subgroup of the compact group ${(G_{d})}^{\land}$ , where $G_{d}$ is the group G with...

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.

Various categorical approaches to statistical spaces

Tadeusz Bromek, Maria Moszyńska

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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

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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...

Displaying similar documents to “Derived functors of lim ← and abelian ab3*- and Ab4*-categories with enough injectives”

Displaying similar documents to “Derived functors of $lim_{\leftarrow}$ and abelian ab3- and Ab4-categories with enough injectives”