范畴M_R~l(Ω)若干性质的研究

研究了Ω-左R-模范畴中的余积及余等值子的性质,揭示了范畴M_R~l(Ω)与范畴M_R~l中余积之间的关系,刻画了范畴M_R~l(Ω)与范畴M_R~l中余等值子之间的关系,同时证明了范畴M_R~l(Ω)的余完备性.     

m-循环复形范畴的Bridgeland-Hall代数的余代数结构

设A是一个遗传Abel范畴且■是A的投射对象构成的满子范畴.本文主要研究胁循环复形范畴C_m(■)的Bridgeland-Hall代数的余代数结构(其中m≥2).受Yanagida工作的启发,我们在C_m(■)上定义一个新的正合结构,由此得到了其Bridgeland-Hall代数的余代数结构.同时,证明了存在A的扩展Ringel-Hall代数到m-循环复形范畴C_m(■)的Bridgeland-Hall代数的余代数嵌入.     

范畴Ω-Cat的极限和余极限

Ω-范畴具有范畴论和序理论的双重意义,可为计算机程序语言的语义提供量化的模型,本文给出了范畴Ω-Cat中极限和余极限的定义,同时研究了范畴Ω-Cat与范畴Set之间极限和余极限的关系。     

Abel范畴的右双-Giraud粘合

我们在本文中引入了Abel范畴的右双-Giraud粘合定义.我们证明了右双-Giraud粘合与余遗传和遗传的挠对存在着双射对应.此外,我们通过模范畴中特定的幂等理想刻画了这类挠对.     

区间值模糊滤子范畴

讨论区间值模糊滤子范畴IVF和Hausdorff区间值模糊滤子范畴IVFHau中乘积、余积、等子、余等子存在性及构造。证明IVF和IVFHau都不是完备范畴而IVF是余完备范畴。也证明IVF不仅有许多弱拓扑满子范畴,也有许多非弱拓扑满子范畴。     

Quantic格范畴

系统研究了Quantic格范畴。证明了Quantic格范畴有等子、余等子。给出了Quantic格范畴中的极限和逆极限结构,从而说明了Quantic格范畴是完备范畴。     

正合范畴中的复形、余挠对及粘合

作者在弱幂等完备的正合范畴(A,E)中引入了复形的新的定义,并且证明了E-正合复形的同伦范畴Kex(E)是同伦范畴KE(A)的厚子范畴.给定(A,E)中的余挠对(x,y),定义了正合范畴(CE(A),C(E))中的两个余挠对((x)E,dg(y)E)和(dg(x)E,(y)E),并且证明了当A是可数完备时,CE(A)中...     

关于余挠三元组的余星子范畴和余倾斜子范畴

设$\mathcal{A}$ 是一个Abel范畴,且 $(\mathcal{X}, \mathcal{Z},\mathcal{Y})$ 是一个完全遗传余挠三元组.介绍 $\mathcal{A}$ 的 $n$-$\mathcal{Y}$-余倾斜子范畴的定义,并给出 $n$-$\mathcal{Y}$-余倾斜子范畴的一个刻画,类似于 $n$-余倾斜模的 Bazzoni 刻画.作为应用,证明了在一个几乎 Gorenstein 环 $R$ 上, 如果 $\mathcal{GP}$ 是 $n$-$\mathcal{GI}$-余倾斜的, 那么 $R$ 是一个 $n$-Gorenstein 环, 其中 $\mathcal{GP}$ 表示 Gorenstein 投射 $R$-模组成的子范畴且 $\mathcal{GI}$ 表示 Gorenstein 内射 $R$-模组成的子范畴. 进而, 研究 任意环$R$上的$n$-余星子范畴, 以及关于余挠三元组 $(\mathcal{P}, R$-Mod, $\mathcal{I})$ 的 $n$-$\mathcal{I}$-子范畴与 $n$-余星子范畴之间的关系, 其中 $\mathcal{P}$ 表示投射左 $R$-模组成的子范畴且 $\mathcal{I}$ 表示内射左 $R$-模组成的子范畴.     

n-予加法范畴中的弱上积与极限

在Abel范畴的同调代数中,双积和环模的正向系统的正向极限都是重要的基本概念。在那里,由于Abel范畴具有零对象这些概念的讨论就比较简单。本文给出这些概念在n-予加法范畴中的一种推广,我们的困难在于n-予加法范畴只具有终对象。本文的结果表明在n-予加法范畴中这些概念的讨论可以不使用零对象,另一方面也给出在Abel范畴场合中讨论极限时使用零对象的实质的一个解剖。     

相对于g(X)的导出范畴

设A是有足够投射对象的Abel范畴,(x,y)是A中的完备遗传余挠对.本文引入Gorenstein x-导出范畴,记作Dg(x)(A),并且从不同的角度研究D_g(x)(A).当(x,y)取作特殊的余挠对时,得到一些已知的导出范畴,如Gorenstein导出范畴和Gorenstein平坦导出范畴等.通过与g(X)有关的同伦范畴K~(-,gxb)(g(x))描述有界Gorenstein x-导出范畴D~b_(g(x))(A)和有界导出范畴D~b(A),并给出一些三角等价.     

广义Comma范畴的Recollement

本文研究广义Comma范畴上Recollement问题.利用Abel范畴上Recollement及其伴随函子,诱导出广义Comma范畴,并利用比较函子构造出广义Comma范畴上的Recollement.这些结果推广了一般Abel范畴上的Recollement,丰富了Comma范畴研究.     

Categorical Generalization of a Universal Domain

J. Adámek defined SC categories as a categorical generalization of Scott domains. Namely, an SC category is finitely accessible, has an initial object and is boundedly cocomplete (each diagram with a compatible cocone has a colimit). SC categories are proved to serve well as a basis for the computer language semantics.The purpose of this paper is to generalize the concept of a universal Scott domain to a universal SC category. We axiomatize properties of subcategories of finitely presentable objects of SC categories (generalizing thus semilattices of compact elements of Scott domains). The categories arising are called FCC (finitely consistently cocomplete) categories. It is shown that there exists a universal FCC category, i.e., such that every FCC category may be FCC embedded into it. The result is an application of a general procedure introduced 30 years ago by V. Trnková.     

Axiomatizing subcategories of Abelian categories

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find 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.     

Relative Regular Objects in Categories

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely generated Grothendieck categories. Applications are given for categories of comodules over a coalgebra and for categories of graded modules, and a link to the theory of generalized inverses of matrices is presented. Some of the techniques we use are new, since dealing with arbitrary categories allows us to pass to the dual category.      

Endomorphism category of an abelian category

Let 𝒞 be an additive category. Denote by End(𝒞) the endomorphism category of 𝒞, i.e., the objects in End(𝒞) are pairs (C,c) with C∈𝒞,c∈End_𝒞(C), and a morphism f:(C,c)→(D,d) is a morphism f∈Hom_𝒞(C,D) satisfying fc?=?df. This paper is devoted to an approach of the general theory of the endomorphism category of an arbitrary additive category. It is proved that the endomorphism category of an abelian category is again abelian with an induced structure without nontrivial projective or injective objects. Furthermore, the endomorphism category of any nontrivial abelian category is nonsemisimple and of infinite representation type. As an application, we show that two unital rings are Morita equivalent if and only if the endomorphism categories of their module categories are equivalent.     

Functorial Calculus in Monoidal Bicategories

The definition and calculus of extraordinary natural transformations is extended to a context internal to any autonomous monoidal bicategory. The original calculus is recaptured from the geometry of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal category V and whose morphisms are modules.     

Properties of abelian categories via recollements

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.     

Subtractive Categories and Extended Subtractions

We introduce a notion of an extended operation which should serve as a new tool for the study of categories like Mal’tsev, unital, strongly unital and subtractive categories. However, in the present paper we are only concerned with subtractive categories, and accordingly, most of the time we will deal with extended subtractions, which are particular instances of extended operations. We show that these extended subtractions provide new conceptual characterizations of subtractive categories and moreover, they give an enlarged “algebraic tool” for working in a subtractive category—we demonstrate this by using them to describe the construction of associated abelian objects in regular subtractive categories with finite colimits. Also, the definition and some basic properties of abelian objects in a general subtractive category is given for the first time in the present paper. The second author acknowledges the support of Claude Leon Foundation, INTAS (06-1000017-8609) and Georgian National Science Foundation (GNSF/ST06/3-004).     

The Left and Right Triangulated Structures of Stable Categories

Beligiannis and Marmaridis in 1994, constructed the one-sided triangulated structures on the stable categories of additive categories induced from some homologically finite subcategories. We extend their results to slightly more general settings. As an application of our results, we give some new examples of one-sided triangulated categories arising from abelian model categories. An interesting outcome is that we can describe the pretriangulated structures of the homotopy categories of abelian model categories via those of stable categories.     

Relative derived equivalences and relative homological dimensions

Let A be a small abelian category. For a closed subbifunctor F of Ext_A¹(-,-), Buan has generalized the construction of Verdier's quotient category to get a relative derived category, where he localized with respect to F-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relative derived categories, we give a relative version on derived equivalences induced by F-tilting complexes. We discuss the relationships between relative homological dimensions and relative derived equivalences.