Categorical Closure Operators

By (author) Castellini, Gabriele

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.

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  • 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

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書名 Categorical Closure Operators
著作者等 Castellini, Gabriele
シリーズ名 Mathematics: Theory and Applications
出版元 Springer-Verlag New York Inc.
刊行年月 2012.10.06
版表示 Softcover reprint of the original 1st ed. 2003
ページ数 316p
大きさ H234 x W156
ISBN 9781461265047
言語 英語
出版国 アメリカ合衆国

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