Teimuraz Pirashvili
Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. He posses also a question whether any abelian category comes in this way. We will give a rather trivial solution of this problem in the case when a given abelian category has enough projective or injective objects.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the transfer of (dual) relative Rickart properties via functors between abelian categories, and we deduce the transfer of (dual) relative Baer property. We also give applications to Grothendieck categories, comodule categories and (graded) module categories, with emphasis on endomorphism rings.
In this paper, we study the relationship of Gorenstein projective objects among three Abelian categories in a recollement. As an application, we introduce the relation of $n$-Gorenstein tilting modules (and Gorenstein syzygy modules) in three Abelian categories. For a recollement of Abelian categories, we show that a resolving subcategory induce two resolving subcategories. On the other hand, we also prove that two resolving subcategories can induce a resolving subcategory. Moreover, we give the size relationship between the relative global dimensions of three Abelian categories.
In this paper, we show all k-linear abelian 1-Calabi-Yau categories over an algebraically closed field k are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of k[[t]]. Since all abelian categories derived equivalent with these two are known, we obtain a classification of all k-linear abelian 1-Calabi-Yau categories up to equivalence.
We give a classification of nullity classes (or torsion classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda's classification of Serre subcategories.
In the efforts to define a 2-categorical analog of an abelian category, two (or three) notions of "abelian 2-categories" are defined. One is the relatively exact 2-category, and the other(s) is the (2-)abelian Gpd-category. We compare these notions. Since they proceed independently in their own way, in different settings and terminologies, it will be worth while to collect and unify them. In this paper, by comparing their definitions and arguments, we show the relationship among these classes of 2-categories.
Galois Connections and Preradicals in Abelian Categories
本文在任意阿贝尔范畴中定义预根运算，证明伴随对诱导伽罗瓦联络，构造α、ω预根，并研究其在双完备及局部小条件下的保持性与格结构。
Basing ourselves on Janelidze and Kelly's general notion of central extension, we study universal central extensions in the context of semi-abelian categories. We consider a new fundamental condition on composition of central extensions and give examples of categories which do, or do not, satisfy this condition.
We study abelian group objects in $ω$-categories and discuss the well-known Dold-Kan correspondence from the perspective of $ω$-categories as a model for strict $\infty$-categories. The first part of the paper is intended to compile results from the existing literature and to fill some gaps therein. We go on to consider a parameterized Dold-Kan correspondence, i.e. a Dold-Kan correspondence for presheaves of $ω$-categories. The main result is to describe the descent or sheaf condition in terms of a glueing condition that is familiar for 1 and 2-stacks.
Spectral Sequences in Semi-Abelian Categories
本文在半阿贝尔范畴中定义谱序列概念，通过双单纯对象构造精确对并导出谱序列，将Quillen的双单纯群结果推广到所有半阿贝尔范畴。
In this paper we define Gabriel monads as the idempotent monads associated to exact reflective localizations in Abelian categories and characterize them by a simple set of properties. The coimage of a Gabriel monad is a Serre quotient category. The Gabriel monad induces an equivalence between its coimage and its image, the localizing subcategory of local objects.
The goal of this thesis is to define a 2-dimensional version of abelian categories, where symmetric 2-groups play the role that abelian groups played in 1-dimensional algebra. Abelian and 2-abelian groupoid enriched categories are defined and it is proved that homology can be developed in them, including the existence of a long exact sequence of homology corresponding to an extension of chain complexes. This generalises known results for symmetric 2-groups. The examples include, in addition to symmetric 2-groups, the 2-modules on a 2-ring, which form a 2-abelian groupoid enriched category. Mor…
Pre-Hilbert $*$-categories: The Hilbert-space analogue of abelian categories
该研究提出准希尔伯特$*$-范畴作为阿贝尔范畴的希尔伯特空间类似物，基于对合反变自函子等结构推导其加性、拟阿贝尔等同调性质，并给出多个相关范畴实例。
Lax Distributivity and a Characterization of Abelian Categories
研究阿贝尔范畴的特征，核心方法是通过余宽松代数、宽松代数及宽松混合重写规则，主要贡献是给出了阿贝尔范畴基于此的一种刻画及相关理论。
A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. In this article we first determine an abstract characterization of the categories which are semi-localizations of an exact Mal'tsev category, by specializing a result due to S. Mantovani. We then turn our attention to semi-abelian categories, where a special type of semi-localizations are known to coincide with torsion-free subcategories. A new characterisation of protomodular categories in terms of binary relations is obtained, inspired by the one discovered in the point…
We study the notion of fundamental group in the framework of descent-exact homological categories. This setting is sufficiently wide to include several categories of "algebraic" nature such as the almost abelian categories, the semi-abelian categories, and the categories of topological semi-abelian algebras. For many adjunctions in this context, the fundamental groups are described by generalised Brown-Ellis-Hopf formulae for the integral homology of groups.
Classes of Universal Epi- and Monomorphisms in Quasi-Abelian Categories
本文解决了一个开放性问题，并提供了非严格 universal epimorphisms 和 monomorphisms 的例子。
There are well known identities that involve the Ext bifunctor, coproducts, and products in Ab4 and Ab4* abelian categories with enough projectives and enough injectives. Namely, for every such category $\mathcal{A}$, the isomorphisms $\operatorname{Ext}^n (\bigoplus_{i\in I}A_{i},X) \cong \prod_{i\in I} \operatorname{Ext}^n(A_{i},X)$ and $\operatorname{Ext}^n (X,\prod_{i\in I}A_{i}) \cong \prod_{i\in I}\operatorname{Ext}^n (X,A_{i})$ always exist. The goal of this paper is to show similar isomorphisms for the Yoneda Ext in Ab4 and Ab4* abelian categories with not necessarily enough projective…
We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi--locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi--locally presentable and we obtain a new proof of Brown representability theorem. Examples of functors which are not representable are also given.
The Category of Hypergroups: seeking for a generalized version of Abelian Categories
本文研究阿贝尔超群范畴的范畴性质，提出超（几乎）预加性和超（几乎）阿贝尔范畴概念，旨在为超代数建立同调代数一般理论，属首次尝试。
This paper is a continuation of our study of non-abelian Baues-Wirsching cohomologies. In our previous paper, we defined second non-abelian cohomology H2(C;D) of a small category C with coefficients in a so-called centralised natural system D. We proved that H2(C;D) classifies linear extensions of C by D, generalising the corresponding result for abelian natural systems. For an abelian natural system D, the third cohomology classifies certain abelian track categories. A track category is a 2-category where all 2-morphisms are isomorphisms. A track category is called abelian if for every 1-morp…
This is an extended version of my earlier articel "Projective and injective objects in symmetric categorical groups. arXiv:1007.0121v1." Several new facts added, including the material on the derived 2-functors and the proof of the Gabriel-Mitchel theorem and the Morita theory for 2-rings.
A well-known theorem of P. Hall, usually called Hall's criterion for nilpotence, states: a group G is nilpotent whenever it has a normal subgroup N such that G/[N,N] and N are nilpotent. We widely generalize this result, replacing groups with objects in an abstract semi-abelian category satisfying suitable conditions. In particular, these conditions are satisfied in any algebraically coherent semi-abelian category, and hence in many categories of classical algebraic structures (including a few where Hall's theorem is already known). Note that the categories of crossed modules, crossed Lie alge…
We study the relation between Bourn's notion of peri-abelian category and conditions involving the coincidence of the Smith, Huq and Higgins commutators. In particular we show that a semi-abelian category is peri-abelian if and only if for each normal subobject $K\leq X$, the Higgins commutator of $K$ with itself coincides with the normalisation of the Smith commutator of the denormalisation of $K$ with itself. We show that if a category is peri-abelian, then the condition (UCE), which was introduced and studied by Casas and the second author, holds for that category. In addition we show, usin…
We define torsion pairs for quasi-abelian categories and give several characterisations. We show that many of the torsion theoretic concepts translate from abelian categories to quasi-abelian categories. As an application, we generalise the recently defined algebraic Harder-Narasimhan filtrations to quasi-abelian categories.
We prove that negative K-groups of small abelian categories are trivial.
We prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg-Moore category of a monad on the category of sets preserve regular epimorphisms.
Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of absolutely pure group valued functors over $\mathcal{M}$, denote by $\mathcal{L}_2(\mathcal{M},\mathcal{G})$, is an abelian category and $\mathcal{M}$ is equivalent to a full subcategory of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ in such a way that $n$-kernels and $n$-cokernels are precisely exact sequences of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ with terms in $\mathcal{M}$. This gives a higher-dimensional version of the Freyd-Mitchell embedding theorem for $n$-abelian categories.
Ternary semi-direct products in semi-abelian categories
本文在半阿贝尔范畴中定义并研究了三元半直积，证明其由满足相容条件的作用和等变态射决定，并探讨了在特定范畴中的简化及与群、李代数版本的关联。
We put cluster tilting in ageneral framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an abelian structure. These abelian quotients turn out to be module categories of Goreisten algebras of dimension at most one.
As a generalization of a Calabi-Yau category, we will say a k-linear Hom-finite triangulated category is fractionally Calabi-Yau if it admits a Serre functor S and there is an n > 0 with S^n = [m]. An abelian category will be called fractionally Calabi-Yau is its bounded derived category is. We provide a classification up to derived equivalence of abelian hereditary fractionally Calabi-Yau categories (for algebraically closed k). They are: the category of finite dimensional representations of a Dynkin quiver, the category of finite dimensional nilpotent representations of a cycle, and the cate…
We study connections between additive and abelian 2-representations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell 2-representation. We also describe, in detail, examples of fiat 2-categories associated to $\mathfrak{sl}_2$-categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur algebras.
We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence.
We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find intrinsic axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any $d$-abelian category is equivalent to a $d$-cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated.
Semi-abelian and finitely cocomplete homological categories are characterized in terms of four resp. three simple axioms, in terms of the basic categorical notions introduced in the first few chapters of MacLane's classical book. As an immediate application we show that categories of diagrams in semi-abelian and similar categories are of the same type; in particular, the category of simplicial or Γ-objects in a semi-abelian category is semi-abelian.
We introduce $n$-abelian and $n$-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that $n$-cluster-tilting subcategories of abelian (resp. exact) categories are $n$-abelian (resp. $n$-exact). These results allow to construct several examples of $n$-abelian and $n$-exact categories. Conversely, we prove that $n$-abelian categories satisfying certain mild assumptions can be realized as $n$-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable categ…
Higher-dimensional generalization of abelian categories via DG-categories
本文提出阿贝尔n-截断DG范畴作为阿贝尔范畴的高维推广，统一了普通阿贝尔范畴与稳定DG范畴，并证明其同伦范畴具有外三角和预三角结构，同时建立了满-单分解等经典理论的高维类比。
The purpose of this article is to study the existence of Deligne's tensor product of abelian categories by comparing it with the well-known ten- sor product of finitely cocomplete categories. The main result states that the former exists precisely when the latter is an abelian category, and moreover in this case both tensor products coincide. An example of two abelian categories whose Deligne tensor product does not exist is given.
We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples.
We study the transfer of (dual) relative CS-Rickart properties via functors between abelian categories. We consider fully faithful functors as well as adjoint pairs of functors. We give several applications to Grothendieck categories and, in particular, to (graded) module and comodule categories.
Stability Conditions on Abelian Comma Categories
该研究针对逗号范畴，给出其为阿贝尔范畴的充分条件，计算其Grothendieck群并给出诺特、阿廷性的充要判据，还定义了阿贝尔逗号范畴上的稳定性条件并实现反向诱导，丰富了范畴论相关理论。
We introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories. Our theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories.
We describe (braided-)commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with finite number of simple local modules.
The stable module category of a selfinjective algebra is triangulated, but need not have any nontrivial $t$-structures, and in particular, full abelian subcategories need not arise as hearts of a $t$-structure. The purpose of this paper is to investigate full abelian subcategories of triangulated categories whose exact structures are related, and more precisely, to explore relations between invariants of finite-dimensional selfinjective algebras and full abelian subcategories of their stable module categories.
For a recollement $(\mathcal{A},\mathcal{B},\mathcal{C})$ of abelian categories, we show that torsion pairs in $\mathcal{A}$ and $\mathcal{C}$ can induce torsion pairs in $\mathcal{B}$; and the converse holds true under certain conditions.
We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent abelian categories and coherent morphisms. These categories link algebra, model theory and "geometry".
We extend to semi-abelian categories the notion of characteristic subobject, which is widely used in group theory and in the theory of Lie algebras. Moreover, we show that many of the classical properties of characteristic subgroups of a group hold in the general semi-abelian context, or in stronger ones.
In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations and we construct several equivalences between deformation problems.
This paper has been withdrawn by the author, since the result was already known.
We introduce and develop the notion of scalar extension for abelian categories. Given a field extension F'/F, to every F-linear abelian category A satisfying a suitable finiteness condition we associate an F'-linear abelian category A' and an exact F-linear functor t: A --> A'. This functor is universal among F-linear right exact functors with target an F'-linear abelian category. We discuss various basic properties of this concept, among others compatibilities with multilinear endofunctors such as tensor products, and the permanence of favourable properties of the functors and categories invo…
We introduce (dual) relative CS-Rickart objects in abelian categories, as common generalizations of (dual) relative Rickart objects and extending (lifting) objects. We study direct summands and (co)products of (dual) relative CS-Rickart objects as well as classes all of whose objects are (dual) self-CS-Rickart. Applications are given to Grothendieck categories and, in particular, to module and comodule categories.
Ladders of recollements of abelian categories are introduced, and used to address three general problems. Ladders of a certain height allow to construct recollements of triangulated categories, involving derived categories and singularity categories, from abelian ones. Ladders also allow to tilt abelian recollements, and ladders guarantee that properties like Gorenstein projective or injective are preserved by some functors in abelian recollements. Breaking symmetry is crucial in developing this theory.
Let $\mathcal{M}$ be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in $\mathcal{M}$. This is based on recent work of Holm and Jørgensen on cotorsion pairs in categories of quiver representations. An application on Ding projective and Ding injective representations of quivers over Ding-Chen rings is given.
In this paper we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular case we also obtain the corresponding obstruction theory for lifting of objects in terms of Yoneda Ext-groups. In appendix we prove the existence of miniversal derived deformations of complexes.
This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild cohomology coincides with the one defined by Gerstenhaber, Schack and Swan in the case of module categories over diagrams and schemes and also with the Hochschild cohomology for exact categories introduced recently by Keller. In addition we show in com…
Abelian envelopes of exact categories and highest weight categories
该研究在正合范畴中定义容许子范畴与薄正合范畴理论，引入正合范畴的左右阿贝尔包络并证明其存在条件，揭示最高权范畴与薄范畴的对应关系，并用导出范畴和Serre函子定义薄正合范畴的对偶，解释Ringel对偶的本质。
In an ongoing project to classify all hereditary abelian categories, we provide a classification of Ext-finite directed hereditary abelian categories satisfying Serre duality up to derived equivalence. In order to prove the classification, we will study the shapes of the Auslander-Reiten components extensively and use appropriate generalisations of tilting objects and coordinates, namely partial tilting sets and probing of objects by quasi-simples.
A functorial approach to $0$-abelian categories
本文用函子方法定义$0$-阿贝尔范畴，构造类似扩张双函子并证明长正合序列等结果，进而研究稳定范畴，并建立其与半遗传环的对应。
Representations over diagrams of abelian categories II: Abelian model structures
本文研究阿贝尔范畴图上的表示，在特定条件下将相容阿贝尔模型范畴族合并为表示范畴上的阿贝尔模型结构，并构造Gorenstein内射与平坦模型结构。
Stratifications of abelian categories
本文研究可通过迭代重摄分解的阿贝尔范畴即分层，给出其等价于有限维代数有限维模范畴及为ε-分层范畴的充要条件，推广了相关已有结论。
We investigate relative CS-Baer objects in abelian categories in relationship with other relevant classes of objects such as relative Baer objects, extending objects, objects having certain summand intersection properties and relative CS-Rickart objects. Dual results are automatically obtained by applying the duality principle in abelian categories. We also study direct sums of relative CS-Baer objects, and we determine the complete structure of dual self-CS-Baer modules over Dedekind domains. Further applications are given to module categories.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。