Chu duality theory and coalgebraic representation of quantum symmetries

Building upon Vaughan Pratt's work on applications of Chu space theory to Stone duality, we develop a general theory of categorical dualities on the basis of Chu space theory and closure conditions, which encompasses a variety of dualities for topological spaces, convex spaces, closure spaces, and measurable spaces (some of which are new duality results on their own). It works as a general method to generate analogues of categorical dualities between frames (locales) and topological spaces beyond topology, e.g., for measurable spaces, convex spaces, and closure spaces. After establishing the Chu duality theory, we apply the state-observable duality between quantum lattices and closure spaces to coalgebraic representations of quantum symmetries, showing that the quantum symmetry groupoid fully embeds into a purely coalgebraic category, i.e., the category of Born coalgebras, which refines, through the quantum duality that follows from Chu duality theory, Samson Abramsky's fibred coalgebraic representations of quantum symmetries (which, in turn, builds upon his Chu representations of symmetries).     

Cotorsion pairs, model category structures, and representation theory

We make a general study of Quillen model structures on abelian categories. We show that they are closely related to cotorsion pairs, which were introduced by Salce [Sal79] and have been much studied recently by Enochs and coauthors [EJ00]. This gives a method of constructing model structures on abelian categories, which we illustrate by building two model structures on the category of modules over a (possibly noncommutative) Gorenstein ring. The homotopy category of these model structures is a generalization of the stable module category much used in modular representation theory. This stable module category has also been studied by Benson [Ben97]. Received: 14 December 2000; in final form: 17 December 2001 / Published online: 5 September 2002     

连续信息基

D.Scott在20世纪60年代引入了domain理论,并给出了两种等价形式——信息系统和邻域系。在型论(typetheory)基础上,G.Sambin提出了形式拓扑(Formal topology)理论,并证明了一元形式拓扑与代数Scott domain等价,从而推出代数信息基的概念,同时作为信息系统与邻域系的推广。本文从信息基的观点出发,提出了连续信息基的概念,证明了它与连续Scott domain的等价性。     

Interpolation categories for homology theories

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions, we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolution model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams-Atiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As an application, we establish an isomorphism between certain E(n)-local Picard groups and some Ext-groups.     

Computable limits and colimits in categories of partial enumerated sets

Computable limits and colimits are “recursive counterparts” of the suitable classical concepts from category theory. We present mainly some interesting problems related to computable products. Moreover, some “computable counterparts” of well-known classical facts from category theory are given. MSC: 03D45, 18A30.     

<Emphasis Type="Italic">G</Emphasis>-manifolds of nilpotent groups and manifolds of power groups

The work is devoted to the theory of manifolds in the category of G-groups and in the category of power groups. Nilpotent manifolds in these categories are studied. Some new notions in group theory are introduced and studied: G-manifolds, groups of reduced G-identities, commutator subgroups, etc. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 46, Algebra, 2007.     

Combinatorial and accessible weak model categories

In a previous work, we have introduced a weakening of Quillen model categories called weak model categories. They still allow all the usual constructions of model category theory, but are easier to construct and are in some sense better behaved. In this paper we continue to develop their general theory by introducing combinatorial and accessible weak model categories. We give simple necessary and sufficient conditions under which such a weak model category can be extended into a left and/or right semi-model category. As an application, we recover Cisinski-Olschok theory and generalize it to weak and semi-model categories. We also provide general existence theorems for both left and right Bousfield localization of combinatorial and accessible weak model structures, which combined with the results above gives existence results for left and right Bousfield localization of combinatorial and accessible left and right semi-model categories, generalizing previous results of Barwick. Surprisingly, we show that any left or right Bousfield localization of an accessible or combinatorial Quillen model category always exists, without properness assumptions, and is simultaneously both a left and a right semi-model category, without necessarily being a Quillen model category itself.     

态射集中的加权星序和加权广义逆

本文推广范畴上态射集中的星序理论.引进范畴上态射集中的加权星序.利用态射集的加权广义逆来刻划态射集中的加权星序.给出了态射集中的加权星序和态射的加权广义逆之间的一些关系.得到了态射集中的加权星序的一些刻划和性质.     

Natural multitransformations of multifunctors

We continue to develop the theory of multicategories over verbal categories. This theory includes both the usual category theory and the theory of operads, as well as a significant part of classical universal algebra. We introduce the notion of a natural multitransformation of multifunctors, owing to which categories of multifunctors from a multicategory to some other one turn into multicategories. In particular, any algebraic variety over a multicategory possesses a natural structure of a multicategory. Furthermore, we construct a multicategory analog of commacategories with properties similar to those in the category case. We define the notion of the center of a multicategory and show that centers of multicategories are commutative operads (introduced by us earlier) and only they. We prove that commutative FSet-operads coincide with commutative algebraic theories.     

GRADED RINGS OF LOCAL FINITE REPRESENTATION TYPE

Abstract

Graded Artin algebras whose category of graded modules is locally of finite representation type are introduced. The representation theory of such algebras is studied. In the hereditary case and in the stably equivalent to hereditary case, such algebras are classified.     

Cartan determinants for gentle algebras

The determinant of the Cartan matrix of a finite dimensional algebra is an invariant of the derived category and can be very helpful for derived equivalence classifications. In this paper we determine the determinants of the Cartan matrices for all gentle algebras. This is a class of algebras of tame representation type which occurs naturally in various places in representation theory. The definition of these algebras is of a purely combinatorial nature, and so are our formulae for the Cartan determinants.Received: 29 October 2004     

Homotopy theory of associative rings

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that—similar to spaces—the derived category obtained by inverting the quasi-isomorphisms is naturally left triangulated. Also, homology theories on rings are studied. These must be homotopy invariant in the algebraic sense, meet the Mayer-Vietoris property and plus some minor natural axioms. To any functor X from rings to pointed simplicial sets a homology theory is associated in a natural way. If X=GL and fibrations are the GL-fibrations, one recovers Karoubi-Villamayor's functors KVi, i>0. If X is Quillen's K-theory functor and fibrations are the surjective homomorphisms, one recovers the (non-negative) homotopy K-theory in the sense of Weibel. Technical tools we use are the homotopy information for the category of simplicial functors on rings and the Bousfield localization theory for model categories. The machinery developed in the paper also allows to give another definition for the triangulated category kk constructed by Cortiñas and Thom [G. Cortiñas, A. Thom, Bivariant algebraic K-theory, preprint, math.KT/0603531]. The latter category is an algebraic analog for triangulated structures on operator algebras used in Kasparov's KK-theory.     

ON EPIDENSE SUBCATEGORIES OF TOPOLOGICAL CATEGORIES

Dense subcategories were introduced by S. Marde?i? for an inverse system approach to (categorical) shape theory.

In this paper some internal characterizations of (epi,bi)dense subcategories of a topological category are given. We also show that if K ? A is a bidense subcategory then the “best approximation” of an A-object X by a K-inverse system is obtained by “modifications” of the structure of X.     

Components of the Fundamental Category

In this article we study the fundamental category (Goubault and Raussen, 2002; Goubault, 2000) of a partially ordered topological space (Nachbin, 1965; Johnstone, 1982), as arising in, e.g., concurrency theory (Fajstrup et al., 1999). The algebra of dipaths modulo dihomotopy (the fundamental category) of such a po-space is essentially finite in a number of situations: We define a component category of a category of fractions with respect to a suitable system, which contains all relevant information. Furthermore, some of these simpler invariants are conjectured to also satisfy some form of a van Kampen theorem, as the fundamental category does (Goubault, 2002; Grandis, 2001). We end up by giving some hints about how to carry out some computations in simple cases.     

Z-Quantale及其范畴性质

本文把集系统的概念应用到Quantale理论中,作为Quantale的一般化,引入了Z-quantale的概念,研究了Z-quantale及其范畴的若干性质.主要结果有:证明了Z-quantale范畴是序半群范畴的反射子范畴,凝聚Z-quantale范畴是Z-quantale范畴的余反射子范畴.讨论了Z-quantale范畴中的投射对象,证明了Z-quantale A是E-投射的当且仅当它是稳定Z-连续的.     

Nearness approximation space based on axiomatic fuzzy sets

The approximation space model was originally proposed by Pawlak (1982) [19]. It was Or?owska who first observed that approximation spaces serves as a formal counterpart of perception, or observation [16, §2, p. 8], in which approximations provide a means of approximating one set of objects with another set of objects using the indiscernibility relation. Topology has been used to enrich the original model of an approximation space as well as more recent models of generalized approximation spaces. In this paper, an extension of th e topology neighborhood based on AFS (Axiomatic Fuzzy Sets) theory is introduced, and some interesting properties are given. Furthermore, a new generalized approximation space model is established with two application examples, which can be used to deal with information tables with many category features and viewed as a multi-granulations form of nearness approximation space models.     

Modules over motivic cohomology

For fields of characteristic zero, we show that the homotopy category of modules over the motivic ring spectrum representing motivic cohomology is equivalent to Voevodsky's big category of motives. The proof makes use of some highly structured models for motivic stable homotopy theory, motivic Spanier-Whitehead duality, the homotopy theories of motivic functors and of motivic spaces with transfers as introduced from ground up in this paper. Working with rational coefficients, we extend the equivalence for fields of characteristic zero to all perfect fields by employing the techniques of alterations and homotopy purity in motivic homotopy theory.     

Algebras of Ehresmann semigroups and categories

E-Ehresmann semigroups are a commonly studied generalization of inverse semigroups. They are closely related to Ehresmann categories in the same way that inverse semigroups are related to inductive groupoids. We prove that under some finiteness condition, the semigroup algebra of an E-Ehresmann semigroup is isomorphic to the category algebra of the corresponding Ehresmann category. This generalizes a result of Steinberg who proved this isomorphism for inverse semigroups and inductive groupoids and a result of Guo and Chen who proved it for ample semigroups. We also characterize E-Ehresmann semigroups whose corresponding Ehresmann category is an EI-category and give some natural examples.     

Bounded Solutions of n-Cocycle And Related Equations on Amenable Semigroups

The cocycle functional equation, originating in group theory and playing a role in such areas as cohomology, polyhedral algebra, and information theory, has a long and rich history. Cocycles of higher orders have been introduced in cohomology theory. This paper presents the bounded solutions of cocycle equations of all orders on amenable semigroups. Some related functional equations are treated also. These results generalize some recent results of Pales and Szekelyhidi.