Another Approach to Topological Descent Theory

In the category T op of topological spaces and continuous functions, we prove that surjective maps which are descent morphisms with respect to the class E of continuous bijections are exactly the descent morphisms, providing a new characterization of the latter in terms of subfibrations E(X) of the basic fibration given by T op/X which are, essentially, complete lattices. Also effective descent morphisms are characterized in terms of effective morphisms with respect to continuous bijections. For classes E satisfying suitable conditions, we show that the class of effective descent morphisms coincides with the one of effective E-descent morphisms.     

Effective Descent Morphisms in Some Quasivarieties of Algebraic, Relational, and More General Structures

We show that the class of effective descent morphisms coincides with the class of regular epimorphisms in suitable categories of internal structures in an exact category. In particular this applies to quasivarieties of (ordinary) first-order structures closed under strong homomorphic images.     

Irreducible Morphisms Between Modules over a Repetitive Algebras

We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra.     

Free Adjunction of Morphisms

We develop a general setting for the treatment of extensions of categories by means of freely adjoined morphisms. To this end, we study what we call composition graphs, i.e. large graphs with a partial binary operation on which we impose only rudimentary requirements. The quasicategory thus obtained contains the quasicategory of all categories as a full reflective subquasicategory; we characterize composition graphs for which this reflexion is of a particularly simple nature.This leads to the concept of semicategory; we apply semicategories to solve characterization problems concerning absolutely initial sources, absolute monosources and potential sections. For instance, we show that in any category, the absolutely initial sources are precisely the sources that contain a section.     

Descent for Regular Epimorphisms in Barr Exact Goursat Categories

We show that the category of regular epimorphisms in a Barr exact Goursat category is almost Barr exact in the sense that (it is a regular category and) every regular epimorphism in it is an effective descent morphism.     

Descent Theory for Schemes

We give a complete characterization of the class of quasi-compact morphisms of schemes that are stable effective descent morphisms for the SCHEMES-indexed category given by quasi-coherent sheaves of modules.     

Effective Descent Morphisms in Categories of Lax Algebras

In this paper we investigate effective descent morphisms in categories of reflexive and transitive lax algebras. We show in particular that open and proper maps are of effective descent, result that extends the corresponding results for the category of topological spaces and continuous maps.     

Free hypermodules: a categorical approach

Abstract

In category theory, the existence of free objects is very important, especially free modules that play an important role in homological algebra. Although algebraic hyperstructures are a natural extension of algebraic structures, due to the major difference between them, study-free objects in algebraic hyperstructures become very difficult. In this article, we provide a categorical approach for the construction of free hypermodules. In fact, by considering appropriate morphisms between hypermodules, we characterize free hypermodules from three different perspectives.     

A note on additive convex processes

We deal with the category $\text{P}$ the objects of which are finite-dimensional, real spaces and the morphisms of which are polyhedral convex processes introduced by R. T. Rockafellar [6]. We give some characterizations of additive processes, and we study superpositions of epimorphisms and monomorphisms in the category $\text{P}$.     

Relative Semi-abelian Categories

We define a relative semi-abelian category as a pair (C,E), where C is a pointed category with finite limits, and E is a class of regular epimorphisms in C satisfying certain conditions, stronger than those defining a relative homological category. Some results on the equivalence of the so-called old-style and new-style axioms for the semi-abelian categories are extended to the relative case. Partially supported by the University of Cape Town Research Associateship and Georgia National Science Foundation (GNSF/ST06/3-004).     

Constellations with range and IS-categories

Constellations are asymmetric generalisations of categories. Although they are not required to possess a notion of range, many natural examples do. These include commonly occurring constellations related to concrete categories (since they model surjective morphisms), and also others arising from quite different sources, including from well-studied classes of semigroups. We show how constellations with a well-behaved range operation are nothing but ordered categories with restrictions. We characterise abstractly those categories that are canonical extensions of constellations with range, as so-called IS-categories. Such categories contain distinguished subcategories of insertions (which are monomorphisms) and surjections (in general different to the epimorphisms) such that each morphism admits a unique factorisation into a surjection followed by an insertion. Most familiar concrete categories are IS-categories, and we show how some of the well-known properties of these categories arise from the fact that they are IS-categories. For appropriate choices of morphisms in each, the category of IS-categories is shown to be equivalent to the category of constellations with range.     

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.     

关于具有泛分解的态射的广义逆

本文给出了预加范畴中具有泛分解的态射的（１,…,ｉ）－逆存在的条件及其表达式．特别地,得到了这类态射的Ｍｏｏｒｅ－Ｐｅｎｒｏｓｅ边和群逆存在的一些新的充要条件和新的表达式．     

A tale of two categories: Inductive groupoids and cross-connections

A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialized groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialized small category whose object set is a strict preorder and the morphisms admit a factorization property. A pair of ‘related’ normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad as categorical models of regular semigroups. We explore the inter-relationship between these seemingly different categorical structures and prove a direct category equivalence between the category of inductive groupoids and the category of cross-connections.     

代数的局部完备集范畴和FS-局部dcpo范畴的笛卡儿闭性

本文引入了代数的局部完备集,FS-局部dcpo,局部稳定映射等概念．主要结果是：以局部Scott连续映射为态射的代数的局部完备集范畴,以局部稳定映射为态射的代数的局部完备集范畴以及以局部Scott连续映射为态射的FS-局部dcpo范畴都是笛卡儿闭范畴．     

Free Factorizations

Given any morphism, we construct extensions of the original category in which this morphism admits certain factorizations, in particular a (retraction, section)-factorization. To this end, we solve the word problem for a certain type of systems of generators and relations for categories. This also enables us to prove preservation properties for the said extensions, e.g. preservation of a pair of diagonalizing classes of epimorphisms and monomorphisms.Iterating such extension processes, we obtain factorizable extensions of categories; in particular, we construct a free proper factorization structure on a given category, which leads to a characterization of preimages of proper factorization structures under full embeddings. As a further application, we characterize an absoluteness property regarding factorizations of functorial images of a morphism.     

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.     

态射的柱心-幂零分解

本文在正合加法范畴中研究了态射的Ｄｒａｚｉｎ逆,证明了态射的柱心－幂零分解的存在性和唯一性．     

Category theory of structural sets with application to mathematical modelling and systems analysis

The extension of N. Bourbaki's structural sets theory is suggested as an effective method for mathematical modelling. Canonical morphisms are introduced for structures of given species. Structural sets and their canonical morphisms constitute the category of a given structure. These constructions permit applications of effective algebraic methods of category theory to analysis of models. The methods developed in the article are applied to the systems theory suggested by author previously. Some results of the systems theory are presented. Applications of the developed methods are demonstrated in examples-a simplified hospital system, for instance.