Ramin Ebrahimi
Our first aim is to provide an analog of the Gabriel-Quillen embedding theorem for $n$-exact categories. Also we give an example of an $n$-exact category that is not an $n$-cluster tilting subcategory, and we suggest two possible ways for realizing $n$-exact categories as $n$-cluster tilting subcategory.
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We introduce the notion of Gabriel filter for a preadditive category C and we show that there is a bijective correspondence between Gabriel filters of C and hereditary torsion theories in the category of additive functors (C,Ab), obtaining a generelization of the theorem given by Gabriel [Ga] and Maranda [Ma] which establishes a bijective correspondence between Gabriel filters for a ring and hereditary torsion theories in the corresponding category of modules.
We compute the Krull-Gabriel dimension of the category of perfect complexes for finite dimensional algebras which are derived discrete.
We develop the theory of 2-quivers and quiver 2-categories to run in parallel with the classical theory of quiver algebras. A quiver 2-category is always finitary, and, conversely, every finitary 2-category will be bi-equivalent with a quiver 2-category for a unique underlying reduced 2-quiver. As an application, we produce an example of a fiat 2-category, one of whose Duflo involutions is not self-adjoint.
It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categori…
On covering theory and its applications
本综述介绍了覆盖技术在 Krull-Gabriel 维数理论中的应用，回顾了箭图与局部有界范畴的经典覆盖理论，展示了函子范畴覆盖理论的最新结果，并重点阐述了这些方法在描述各类代数和局部有界范畴的 Krull-Gabriel 维数中的应用。
Gabriel Spectrum of Persistence Categories
本文用纯拓扑方法确定了向量空间层范畴的 Gabriel 谱，证明其同胚于索伯化空间的 Skula 拓扑，并应用于持久性理论，得到持久模范畴的 Gabriel 谱的具体计算。
On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras
本文研究了 cluster repetitive categories 和 cluster-tilted algebras 的 Krull-Gabriel 维数，证明了其与 repetitive category 维数的不等与相等关系，并在附录中给出了另一种确定维数的方法。
The first Brauer-Thrall conjecture for extriangulated length categories
本文在外三角长度范畴中引入关于 $Θ$ 的 Gabriel-Roiter 测度并推广 Gabriel 性质，在技术性条件下证明第一 Brauer-Thrall 猜想。
For an abelian category $\mathcal{A}$ we investigate when the stable categories $\underline{\mathrm{GPro}}\mathrm{j}(\mathcal{A})$ and $\underline{\mathrm{GIn}}\mathrm{j}(\mathcal{A})$ are triangulated equivalent. To this end, we realize these stable categories as homotopy categories of certain (non-trivial) model categories and give conditions on $\mathcal{A}$ that ensure the existence of a Quillen equivalence between the model categories in question. We also study when such a Quillen equivalence transfers from $\mathcal{A}$ to categories naturally associated to $\mathcal{A}$, such as $\mathr…
Coxeter quiver representations in fusion categories and Gabriel's theorem
本文引入Coxeter箭图表示概念，利用$U_q(\mathfrak{s}\mathfrak{l}_2)$融合范畴推广了经典箭图表示理论，证明了涵盖所有Coxeter--Dynkin图的广义Gabriel定理，并揭示了不可分解表示与正根的双射关系。
Quillen equivalence for chain homotopy categories induced by balanced pairs
研究平衡对诱导的链同伦范畴的三角等价条件，通过模型范畴的同调范畴实现，并应用于余内射态射、Gorenstein射影与内射模及纯射影与纯内射对象。
This paper consists of three results on Frobenius categories: (1) we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an extension-closed exact subcategory of the Frobenius category formed by Cohen-Macaulay modules over some additive category; this is an analogue of Gabriel-Quillen's embedding theorem for Frobenius categories; (3) we show that under certain conditions an exact category with enough projective and enough injective objects allows a natural new exact structure, with …
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 $\underline{\F}$ as triangulated categories.
The resolving completion of an exact category
本文通过导出范畴和Gabriel-Quillen嵌入两种方法，研究精确范畴的解析完成问题，提出预aisle及其右正交的构造，并探讨其作为t结构的充要条件。
We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting in higher cluster categories and especially an algorithm to determine the Gabriel quivers of tilting objects in such categories.
We use Quillen model structures to show a systematic method to lift recollements of hereditary abelian model categories to recollements of their associated homotopy categories. To that end, we use the notion of Quillen adjoint triples and we investigate transfers of abelian model structures along adjoint pairs. Applications include liftings of recollements of module categories to their derived counterpart, liftings to homotopy categories that provide models for stable categories of Gorenstein projective and injective modules and liftings to homotopy categories of n-morphism categories over Iwa…
Separable extensions in tensor-triangular geometry and generalized Quillen stratification
本文分析张量三角范畴可分扩张诱导的谱映射，确定其像与纤维基数，并建立直至幂零的弱下降，从而将 Quillen 分层定理推广至等变导出范畴。
Covering theory, functor categories and the Krull-Gabriel dimension
本文通过发展函子范畴的Galois覆盖理论，证明了局部有界K-范畴与其Galois G-覆盖的Krull-Gabriel维数不等式，引入两类函子并给出相关应用，为Prest猜想研究提供支撑。
With the help of the tools of Quillen's homotopical algebra, we construct `the universal additive invariant', namely a functor from the category of small dg categories to an additive category, that inverts the Morita dg functors, transforms the semi-orthogonal decompositions in the sense of Bondal-Orlov into direct sums and which is universal for these properties. We compare our construction with that of Bondal-Larsen-Lunts. ----- A l'aide des outils de l'algebre homotopique de Quillen, on construit `l'invariant additif universel', c'est-a-dire un foncteur defini sur la categorie des petites d…
On exact dg categories
该研究引入正合dg范畴概念，推广了多种经典正合相关范畴概念，给出类Quillen的同伦范畴替换定义，得到dg神经、导出范畴等方面的基本结果，包括正合结构双射与最大正合结构存在性结论。
We prove nilpotence theorems in tensor-triangulated categories using suitable Gabriel quotients of the module category, and discuss examples.
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