Components of the Fundamental Category II

In this article we carry on the study of the fundamental category (Goubault and Raussen, Dihomotopy as a tool in state space analysis. In: Rajsbaum, S. (ed.) LATIN 2002: Theoretical Informatics. Lecture Notes in Computer Science, vol. 2286, Cancun, Mexico, pp. 16–37, Springer, Berlin Heidelberg New York, 2002; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003) of a partially ordered topological space (Nachbin, Topology and Order, Van Nostrand, Princeton, 1965; Johnstone, Stone Spaces, Cambridge University Press, Cambridge, MA, 1982), as arising in e.g. concurrency theory (Fajstrup et al., Theor. Comp. Sci. 357: 241–278, 2006), initiated in (Fajstrup et al., APCS, 12(1): 81–108, 2004). The “algebra” of dipaths modulo dihomotopy (the fundamental category) of such a po-space is essentially finite in a number of situations. We give new definitions of the component category that are more tractable than the one of Fajstrup et al. (APCS, 12(1): 81–108, 2004), as well as give definitions of future and past component categories, related to the past and future models of Grandis (Theory Appl. Categ., 15(4): 95–146, 2005). The component category is defined as a category of fractions, but it can be shown to be equivalent to a quotient category, much easier to portray. A van Kampen theorem is known to be available on fundamental categories (Grandis, Cahiers Topologie Géom. Différentielle Catég., 44: 281–316, 2003; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003), we show in this paper a similar theorem for component categories (conjectured in Fajstrup et al. (APCS, 12(1): 81–108, 2004). This proves useful for inductively computing the component category in some circumstances, for instance, in the case of simple PV mutual exclusion models (Goubault and Haucourt, A practical application of geometric semantics to static analysis of concurrent programs. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005 – Concurrency Theory: 16th International Conference, San Francisco, USA, August 23–26. Lecture Notes in Computer Science, vol. 3653, pp. 503–517, Springer, Berlin Heidelberg New York, 2005), corresponding to partially ordered subspaces of R ⁿ minus isothetic hyperrectangles. In this last case again, we conjecture (and give some hints) that component categories enjoy some nice adjunction relations directly with the fundamental category.      

拟Abelian范畴上函子范畴的局部化

通过拟Abelian范畴的局部类构造出函子范畴的局部类,进一步研究函子范畴的局部化范畴与局部化范畴的函子范畴之间的关系.     

纯奇点范畴中的Buchweitz定理

我们定义纯奇点范畴D_(psg)~b(R)为有界纯导出范畴D_(pur)~b(R)与纯投射模构成的有界同伦范畴K~b(■)的Verdier商,得到了纯奇点范畴D_(psg)~b(R)三角等价于相对纯投射模的Gorenstein范畴的稳定范畴■的一个充分必要条件.同时,还给出三角等价D_(psg)~b(R)≌D_(psg)~b(S)的充分条件,这里R和S都是环.     

范畴中态射集减序的刻划

本文研究了范畴中态射集减序的刻划问题，其主要结果是：（１）证得一般范畴情形减序的十四种刻划；（２）将Ｍｉｔｒａ、Ｂａｋｓａｌａｒｙ等人的利用（１）－逆、（１，２）－逆的包含关系及平行和来刻划复矩阵减序的结果推广到素性预加法范畴与带对合的正性预加法范畴中；（３）给出了１９８０年Ｈａｒｔｗｉｇ关于半群减序刻划的一个反例，因此否定了这个刻画．     

Internal Categories in a Left Exact Cosimplicial Category

The notion of an internal category in a left exact cosimplicial category is introduced. For any topos over sets a certain left exact cosimplicial category is constructed functorially and the category of internal categories in it is investigated. The notion of a fundamental group is defined for toposes admitting the notion of a discrete category.     

The Category of Directed Systems in a Category

We present two related categorical constructions. Given a category C, we construct a category C^[d], the category of directed systems in C. C embeds into C^[d], and if C has enough colimits, then C is monadic over C^[d]. Also, if E,M is a factorization structure for C, then C^[d] has a related factorization structure E_d M_d such that if E consists entirely of monic arrows, then so does E_d and the E_d-quotient poset of an object A is naturally the poset of directed downsets of the E-quotient poset of A. Similarly, if M consists entirely of monicarrows, then so does M_d and the M_d-subobject poset of an object A is naturally the poset of directed downsets of the M-subobject poset. C^[d] has completeness and cocompleteness properties at least as good as those of C, and it is abelian if C is. Dualization gives the other construction: a category C^[i], the category of inverse systems in C, into which C also embeds and which satisfies similar properties, except that directed downsets in the E-quotient and M-subobject posets are replaced by directed upsets.     

A Note on the Radical of a Module Category

We characterize the finiteness of the representation type of an artin algebra in terms of the behavior of the projective covers and the injective envelopes of the simple modules with respect to the infinite radical of the module category. In case the algebra is representation-finite, we show that the nilpotency of the radical of the module category is the maximal depth of the composites of these maps, which is independent from the maximal length of the indecomposable modules.     

Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups

We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form × D,where h has infinite order and D is finite.     

拉回正合范畴中小子对象的探讨

拉回正合范畴是Abelian范畴的真正推广,是界在正合范畴与Abelian范畴之间的一类范畴.本文利用拉回-推出,引进拉回正合范畴的小子对象概念,并给出小子对象相关的性质以及等价条件.     

On the Fundamental Theorem of Algebra

In the paper, it is shown how, by using the minimal means and without introducing complex numbers, one can prove the fundamental theorem of algebra for real polynomials. The fundamental theorem for polynomials with complex coefficients is an immediate corollary of the theorem mentioned.     

The Ker-Coker-sequence and its generalization in some classes of additive categories

We study the validity of the Snake Lemma (the existence and exactness of the Ker-Cokersequence) in a P-semi-abelian category. We also obtain a generalization of the Snake Lemma in a quasiabelian category.     

Weak Factorizations,Fractions and Homotopies

We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel. Mathematics Subject Classifications (2000) 18A99, 55P10, 68Q85.JiÍ Rosický: Supported by the Grant Agency of the Czech Republic under the grant 201/02/0148.     

加法范畴中态射的双积与群逆

研究了加法范畴上具有满单分解的态射的群逆,通过构造双积给出了群逆存在的充要条件及表达式.     

具有广义分解态射的加权Moore-Penrose逆

研究范畴中态射的加权Moore-Penrose逆,利用态射广义分解的性质给出了态射加权Moore-Penrose逆存在的一些充要条件,导出了态射的加权Moore-Penrose逆的表达式,推广了态射Moore-Penrose逆的相应结果.     

Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.     

关于具有泛分解态射的加权Moore-Penrose逆

给出了预加法范畴中具有泛分解态射的加权Moore-Penrose逆存在的充要条件及其表达式,推广了具有泛分解的态射的Moore-Penrose逆的相应结果.     

Complex eigenvalues before the Fundamental Theorem of Algebra

We present an elementary proof of the existence of an eigenvalue for an endomorphism of a complex vector space and we derive the Fundamental Theorem of Algebra as a corollary of this existence. We also present new proofs for the corresponding results for endomorphisms of real vector spaces.     

余Frame范畴中余积的构造

首先,在并半格中引入了上覆盖关系的概念,并以此为基础引入强并半格以及强并半格中上覆盖和C-滤子的概念,证明了强并半格S中全体C-滤子之族C Fil(S)是余Frame,讨论了简单上集值映射u:S→C Fil(S)的相关并半格同态性质;其次,证明了由一族余Frame{A_λ|λ∈Γ}的直积Π_(λ∈Γ)A_λ中只有有限个坐标非零的元素构成的子集A是强并半格,还证明了A是余Frame族{A_λ|λ∈Γ}在并半格范畴中的余积对象;最后,通过各个坐标集中的上覆盖关系在A中定义了上覆盖C~*,再结合简单上集值映射u:A→C~*Fil(A)和标准入射qλ:Aλ→Π_(λ∈Γ)A_λ(λ∈Γ),证明了强并半格A中由上覆盖C~*诱导的余Frame C~*Fil(A)是余Frame族{A_λ|λ∈Γ}在余Frame范畴中的余积对象.     

关于基本群的非交换性

设{s1∨s2,y0}为具有公共切点y0的两个圆s1、s2的并集,通过对s1∨s2中的每一个闭路径定义一个“点列”,给出基本群π1(s1∨s2,y0)非交换性的一个新的证明.     

简单弧上带Cauchy核的第一类奇异积分方程的新解法

提供了简单弧上带Cauchy核的第一类奇异积分方程1πi∫ltf-(tt)0dt+21πi∫lk(t,t0)f(t)dt=g(t0)的一个新解法,给出了解的具体表达式与运算实例.本文的方法可应用于更广泛的情形.其中k(t,t0)为多项式,t0∈l,t0不为l的端点,f(t)在端点的奇性不到一阶.