张量余单子的余半单性与余辫子结构

本文研究了张量余单子的余半单性和余表示范畴,给出了其余半单性和余可裂性的等价性定理.并证明了其余表示范畴是辫子范畴当且仅当该张量余单子是余辫子的.作为应用研究了张量型Hom-双代教的Hom-余模范畴的半单性和辫子结构.     

双单子分配律的R-矩阵

本文讨论了双单子分配律的表示及其R-矩阵结构.设F和G是给定的双单子,刻画了单子双模范畴,并给出了其为辫子范畴的充要条件,由此构造了量子YangBaxter方程的一组新解系.     

模糊理论与Kleisli范畴

运用范畴论的观点和语言,讨论了几种真值集不同的模糊集,得出它们都是特殊的模糊理论.更进一步,指出了模糊理论所对应的范畴与由模糊理论诱导的单子所构造的Kleisli范畴的等价关系.最后,通过一个实例,描述了伴随函子诱导的单子,并构造了相应的Kleisli范畴,指出了Kleisli范畴在模糊理论中的应用.     

态族带序结构的范畴

范畴的态族可以赋予一些结构,如Ab范畴,但在自然界中我们大量接触的是连续现象,作为现代分析的基础,连续是用序关系处理的.因此,本文赋予范畴的态族一个序结构,引进了带序范畴和正序函子,以此作为连续性的一种范畴描述.如范畴C带T_0拓扑,若(?)U_fg,(?)U_f,U_g,使得U_fg=U_f.U_g.则按开集的一种包含关系即构造出带拓扑的范畴上的序结构,从而在拓     

n-角范畴的局部化

本文研究了n-角范畴的局部化理论.利用给定的n-角范畴K和K的一个相容乘法系S,通过局部化方法构造与原范畴对象相同的商范畴S-1K,在新范畴中S的态射成为同构,并且该商范畴具有n-角结构且满足一定的泛性,推广了三角范畴的局部化理论.     

Yetter-Drinfel'd范畴中辫化余交换余代数

本文研究与Ｈｏｐｆ代数Ｈ关联之ＹｅｔｅｒＤｒｉｎｆｅｌ’ｄ范畴ＹＨＤ中的辫化余交换余代数Ｃ，证明ＨＹＤ中左Ｃ－余模范畴ＨＹＤ是张量范畴，且ＨＹＤ中辫结构Ψ诱导ＣＨＹＤ中一辫结构当且仅当对ＣＨＤＹ中任意对象Ｎ有ΨＮ，ＣΨＣ，ＮＣΓＮ＝ＣＹＤΓＮ；由此导致新的辫化张量范畴．     

稳定范畴与三角等价

外三角范畴是三角范畴和正合范畴的推广,很多重要的结论都能统一在这个框架下.本文通过外三角范畴中的理想商实现了模型结构诱导的局部化商,即建立了这两种商范畴之间的三角等价.与此同时,也将此结果应用到Frobenius正合范畴、环R的同伦范畴和Gorenstein对象的稳定范畴.     

基于群范畴的L-fuzzy结构提升范畴及其表示

引入群范畴上L-fuzzy结构提升范畴与格值结构提升范畴概念,L-fuzz结构是在逻辑层面表达群理论多值语义的有点化描述,也是Zermelo-Fr(a)nkel公理集合理论和各种代数形式理论的格值模型的语义赋值,而格值结构是在范畴层面表达群理论多值语义的无点化描述,本文建立了L-fuzzy结构与格值结构这两种不同数学结构之间的联系,证明了在范畴层面上述两种结构是同构的.给出了基于群范畴的L-fuzzy结构的格值结构表示.     

外三角范畴的同调系统

在外三角范畴中,本文引入同调系统Θ(又称为Θ-系统)的概念,此概念统一了模范畴的分层系统和三角范畴的同调系统.本文证明了一个Θ-系统能够唯一地确定一个Θ-投射系统.给定一个Θ-投射系统(Θ, Q,≤),本文也证明了滤链多样性不依赖于滤链的选择,建立了所有Θ-滤对象构成的子范畴■(Θ)和模范畴mod(B)中的所有?-好模构成的子范畴之间的同构,其中B是标准分层代数.     

三角矩阵代数上奇点范畴和Gorenstein亏范畴的2-粘合

设■是三角矩阵代数,其中A和B是Artin代数,_AM_B是A-B-双模.本文研究了T上奇点范畴和Gorenstein亏范畴的2-粘合结构.在恰当的假设下,我们给出了T上奇点范畴和Gorenstein亏范畴的2-粘合存在的充分必要条件.     

Skew Monoidal Monoids

Skew monoidal categories are monoidal categories with non-invertible “coherence” morphisms. As shown in a previous article, bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod-R in which the unit object is R_R. This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present article, we study the one-object case: skew monoidal monoids (SMMs). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. These bialgebroids are submonoids of their own base and are rank 1 free over the base on the source side. We give various equivalent definitions of SMM, study the structure of their (co)module categories, and discuss the possible closed and Hopf structures on a SMM.     

Monoidal composition of distributions

Summary A useful property of a Poisson process is that if occurrences are independently selected with probability a, then the resulting process is Poisson with mean a, where is the original process mean. This property is examined from an abstract viewpoint under a natural restriction on the selection mechanism, namely that if a, b, characterize two selection mechanisms of interest, then the composite selection, when acting on a given distribution, is characterized by a o b, where o is an associative operation. In the terminology of Bourbaki, the quantities, a,b,..., together with o, form a monoid. The monoid will, for simplicity, be assumed to possess a two-sided unit e. The class of processes is generalized under a related closure restriction, which is that the distributions are members of a parametric family which is invariant under a monoidal selection mechanism. Various consequences of these assumptions are deduced, relating to the form of the selection mechanism and of the parametric families. The methods of Aczél are used here.Under the special assumption that the monoid involved is the multiplicative monoid of reals in the unit interval, that the selection is positively compatible with the parametric family, and that the generating functions are univariate and of Mandelbrojt type, it follows from the Bernstein-Widder Theorem that the distributions are mixtures of stuttering (also called compound) Poisson distributions. Moreover, it is shown that every parametric family is positively compatible with linear selection, and that negative binomial distributions are positively compatible with exponential selection.     

Monoidal structures for N-complexes

We classify the monoidal structures for the category of N-complexes which respect the graded structure.     

Monoidal cut strengthening revisited

There are two distinct strengthening methods for disjunctive cuts with some integer variables; they differ in the way they modularize the coefficients. In this paper, we introduce a new variant of one of these methods, the monoidal cut strengthening procedure, based on the paradox that sometimes weakening a disjunction helps the strengthening procedure and results in sharper cuts. We first derive a general result that applies to cuts from disjunctions with any number of terms. It defines the coefficients of the cut in a way that takes advantage of the option of adding new terms to the disjunction. We then specialize this result to the case of split cuts for mixed integer programs with some binary variables, in particular Gomory mixed integer cuts, and to intersection cuts from multiple rows of a simplex tableau. In both instances we specify the conditions under which the new cuts have smaller coefficients than the cuts obtained by either of the two currently known strengthening procedures.     

Monoidal categories of corings

We introduce a monoidal category of corings using two different notions of corings morphisms. The first one is the (right) coring extensions recently introduced by T. Brzeziński in [2], and the other is the usual notion of morphisms defined in [6] by J. Gómez-Torrecillas.

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Sunto Introduciamo una categoria monoidale di coanelli usando due diverse nozioni di morfismi di coanelli. La prima è l'estensione (destra) di coanelli recentemente introdotta da Brzeziński in [2], mentre la seconda è la nozione usuale di morfismo definita in [6] da J. Gómez-Torrecillas.

     

Monoidal Categories and Multiextensions

We associate to a group-like monoidal grupoid a principal bundle E satisfying most of the axioms defining a biextension. The obstruction to the existence of a genuine biextension structure on E is exhibited. When this obstruction vanishes, the biextension E is alternating and a trivialization of E induces a trivialization of . The analogous theory for monoidal n-categories is also examined, as well as the appropriate generalization of these constructions in a sheaf-theoretic context. In the n-categorial situation, this produces a higher commutator calculus, in which some interesting generalizations of the notion of an alternating biextension occur. For n=2, the corresponding cocycles are constructed explicitly, by a partial symmetrization process, from the cocycle describing the n-category.     

Militaru's D-Equation in Monoidal Categories

A FRT type construction is done in a minimal categorical context: the ambient monoidal category is only assumed to have coequalizers. The early motivation for this construction was G. Militaru's work on the D-equation. We get generalizations of Militaru's constructions and results. The D-equation is also studied using the classical FRT construction: this leads to a notion of D-bialgebra. New solutions of the D-equation are constructed.     

Weak Bialgebras and Monoidal Categories

We study monoidal structures on the category of (co)modules over a weak bialgebra. Results due to Nill and Szlachányi are unified and extended to infinite algebras. We discuss the coalgebra structure on the source and target space of a weak bialgebra.     

Cotensor Coalgebras in Monoidal Categories

We introduce the concept of cotensor coalgebra for a given bicomodule over a coalgebra in an Abelian monoidal category ?. If ? is also cocomplete, complete, and AB5, we show that such a cotensor coalgebra exists and satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration is filled by considering a direct limit of a filtration consisting of wedge products. We prove that this coalgebra is formally smooth whenever the comodule is relative injective and the coalgebra itself is formally smooth.