Categorical Closure Operators. Mathematics: Theory & Applications

Seitenzahl:

300

Gewicht:

637 g

Erschienen bei:

Springer

Beschreibung

This book presents the general theory of categorical closure operators
together with examples and applications to the most common categories,
such as topological spaces, fuzzy topological spaces, groups, and
abelian groups. The main aim of the theory is to develop a categorical
characterization of the classical basic concepts in topology via the
newly introduced concept of categorical closure operators. This
permits many topological ideas to be introduced in a topology-free
environment and imported afterwards into a new category, which often
yields interesting new insights into its structure.
The first part of the book deals with the general theory, starting
with basic definitions and gradually moving to more advanced
properties. The second part includes applications to the classical
concepts of epimorphisms, separation, compactness and connectedness.
Every chapter ends with exercises. A comprehensive list of references
for the reader who wants to consult original works and a good index
complete the book.
"Categorical Closure Operators" is self-contained and can be
considered as a graduate level text for topics courses in category
theory, algebra, and topology. The book appeals mainly to graduate
students and researchers in category theory and categorical topology,
and to those interested in categorical methods applied to the most
common concrete categories. The reader is expected to have some basic
knowledge of algebra, topology and category theory; however, all
recurrent categorical concepts are included in a preliminary chapter.

Kurzbeschreibung

This book presents the general theory of categorical closure operators together with examples and applications to the most common categories, such as topological spaces, fuzzy topological spaces, groups, and abelian groups. The main aim of the theory is to develop a categorical characterization of the classical basic concepts in topology via the newly introduced concept of categorical closure operators. This permits many topological ideas to be introduced in a topology-free environment and imported afterwards into a new category, which often yields interesting new insights into its structure. The first part of the book deals with the general theory, starting with basic definitions and gradually moving to more advanced properties. The second part includes applications to the classical concepts of epimorphisms, separation, compactness and connectedness. Every chapter ends with exercises. A comprehensive list of references for the reader who wants to consult original works and a good index complete the book. "Categorical Closure Operators" is self-contained and can be considered as a graduate level text for topics courses in category theory, algebra, and topology. The book appeals mainly to graduate students and researchers in category theory and categorical topology, and to those interested in categorical methods applied to the most common concrete categories. The reader is expected to have some basic knowledge of algebra, topology and category theory; however, all recurrent categorical concepts are included in a preliminary chapter. TOC:Preface * Introduction * Part I: General Theory * Galois Connections * Some Categorical Concepts * Factorization Structures for Sinks * Closure Operators: Definitions and Examples * Idempotency, Weak Heredity and Factorization Structures * Additivity, Heredity, Suprema and Infima of Closure Operators * Additional Description of \hat{C} and \check{C} and Subobject Orthogonality * A Diagram Of Galois Connections of Closure Operators * Regular Closure Operators * Hereditary Regular Closure Operators * Part II- Applications * Epimorphisms * Separation * Compactness * Connectedness * Connectedness in Categories with a Terminal Object * A Link between two Connectedness Notions * Different Constructions Related * References * List of Symbols * Index

Mehr über...

Mehr über:  Algebra, algebra, topology, category theory, Kategorie - kategorial, kategorial ( Kategorie ), Operatoren

Mehr von: 

Mehr von:  Gabriele Castellini, Springer