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2021 | Book

Ring Epimorphisms, Gabriel Topologies and Contramodules Read chapter Read first chapter

New Perspectives in Algebra, Topology and Categories

Summer School, Louvain-la-Neuve, Belgium, September 12-15, 2018 and September 11-14, 2019

Editors: Prof. Maria Manuel Clementino, Alberto Facchini, Prof. Marino Gran

Publisher: Springer International Publishing

Book Series : Coimbra Mathematical Texts

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book provides an introduction to some key subjects in algebra and topology. It consists of comprehensive texts of some hours courses on the preliminaries for several advanced theories in (categorical) algebra and topology. Often, this kind of presentations is not so easy to find in the literature, where one begins articles by assuming a lot of knowledge in the field. This volume can both help young researchers to quickly get into the subject by offering a kind of « roadmap » and also help master students to be aware of the basics of other research directions in these fields before deciding to specialize in one of them. Furthermore, it can be used by established researchers who need a particular result for their own research and do not want to go through several research papers in order to understand a single proof. Although the chapters can be read as « self-contained » chapters, the authors have tried to coordinate the texts in order to make them complementary.
The seven chapters of this volume correspond to the seven courses taught in two Summer Schools that took place in Louvain-la-Neuve in the frame of the project Fonds d’Appui à l’Internationalisation of the Université catholique de Louvain to strengthen the collaborations with the universities of Coimbra, Padova and Poitiers, within the Coimbra Group.

Table of Contents

Frontmatter
Chapter 1. Ring Epimorphisms, Gabriel Topologies and Contramodules
Abstract
During the 1960s considerable work was done in order to understand the meaning of “epimorphism”. The notion plays an important role in categories of rings where the abstract category-theoretic meaning is now of common use.
The notion of ring epimorphism has relations with torsion theory and localisation theory. In particular, perfect right Gabriel topologies (in Stenström’s terminology) correspond bijectively to left flat ring epimorphisms.
In these notes we will consider two classes of modules defined in terms of a ring epimorphism: the comodules and the contramodules as introduced by Leonid Positselski. Adding mild conditions on the ring epimorphism we will extend classical results proved by Matlis for commutative rings by showing an equivalence between suitable subcategories of the two classes of comodules and contramodules.
Silvana Bazzoni
Chapter 2. An Invitation to Topological Semi-abelian Algebras
Abstract
In this text we present the fundamental results that show that both classical topological properties of topological groups and categorical properties of the category of topological groups and continuous homomorphisms can be extended to the more general setting of topological semi-abelian algebras.
Maria Manuel Clementino
Chapter 3. Commutative Monoids, Noncommutative Rings and Modules
Abstract
These are the notes of a non-standard course of Algebra. It deals with elementary theory of commutative monoids and non-commutative rings. Most of what is taught in a master course of Commutative Algebra holds not only for commutative rings, but more generally for any commutative monoid, which shows that the additive group structure on a commutative ring has little importance.
In the rest of the notes of the course presented here, we introduce the basic notions of non-commutative rings and their modules, stressing the difference with what happens in the case of commutative rings.
Alberto Facchini
Chapter 4. An Introduction to Regular Categories
Abstract
This paper provides a short introduction to the notion of regular category and its use in categorical algebra. We first prove some of its basic properties, and consider some fundamental algebraic examples. We then analyse the algebraic properties of the categories satisfying the additional Mal’tsev axiom, and then the weaker Goursat axiom. These latter contexts can be seen as the categorical counterparts of the properties of 2-permutability and of 3-permutability of congruences in universal algebra. Mal’tsev and Goursat categories have been intensively studied in the last years: we present here some of their basic properties, which are useful to read more advanced texts in categorical algebra.
Marino Gran
Chapter 5. Categorical Commutator Theory
Abstract
In these notes, we introduce the reader to the categorical commutator theory (of subobjects), following the formal approach given by Mantovani and Metere in 2010. Such an approach is developed along the lines provided by Higgins, based on the notion of commutator word, introduced by the author in the context of varieties of \(\Omega \)-groups (groups equipped with additional algebraic operations of signature \(\Omega \)). An internal interpretation of the commutator words is described, providing an intrinsic notion of Higgins commutator, which reveals to have good properties in the context of ideal determined categories. Furthermore, we will illustrate some applications of commutator theory in categorical algebra, such as a useful way to test the normality of subobjects on one side, and the construction of the abelianization functor on the other.
Sandra Mantovani, Andrea Montoli
Chapter 6. Notes on Point-Free Topology
Abstract
Point-free topology is the study of the category of locales and localic maps and its dual category of frames and frame homomorphisms. These notes cover the topics presented by the first author in his course on Frames and Locales at the Summer School in Algebra and Topology. We give an overview of the basic ideas and motivation for point-free topology, explaining the similarities and dissimilarities with the classical setting and stressing some of the new features.
Jorge Picado, Aleš Pultr
Chapter 7. Non-associative Algebras
Abstract
A non-associative algebra over a field \(\mathbb {K}\) is a \(\mathbb {K}\)-vector space A equipped with a bilinear operation \( {A\times A\rightarrow A:\; (x,y)\mapsto x\cdot y=xy}\). The collection of all non-associative algebras over \(\mathbb {K}\), together with the product-preserving linear maps between them, forms a variety of algebras: the category \(\mathsf {Alg}_\mathbb {K}\). The multiplication need not satisfy any additional properties, such as associativity or the existence of a unit. Familiar categories such as the varieties of associative algebras, Lie algebras, etc. may be found as subvarieties of \(\mathsf {Alg}_\mathbb {K}\) by imposing equations, here \(x(yz)=(xy)z\) (associativity) or \(xy =- yx\) and \(x(yz)+z(xy)+ y(zx)=0\) (anti-commutativity and the Jacobi identity), respectively.
The aim of these lectures is to explain some basic notions of categorical algebra from the point of view of non-associative algebras, and vice versa. As a rule, the presence of the vector space structure makes things easier to understand here than in other, less richly structured categories.
We explore concepts like normal subobjects and quotients, coproducts and protomodularity. On the other hand, we discuss the role of (non-associative) polynomials, homogeneous equations, and how additional equations lead to reflective subcategories.
Tim Van der Linden
Metadata
Title
New Perspectives in Algebra, Topology and Categories
Editors
Prof. Maria Manuel Clementino
Alberto Facchini
Prof. Marino Gran
Copyright Year
2021
Electronic ISBN
978-3-030-84319-9
Print ISBN
978-3-030-84318-2
DOI
https://doi.org/10.1007/978-3-030-84319-9

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