Categorical Closure Operators
(Sprache: Englisch)
This book presents the general theory of categorical closure operators to gether with a number of examples, mostly drawn from topology and alge bra, which illustrate the general concepts in several concrete situations. It is aimed mainly at researchers and...
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This book presents the general theory of categorical closure operators to gether with a number of examples, mostly drawn from topology and alge bra, which illustrate the general concepts in several concrete situations. It is aimed mainly at researchers and graduate students in the area of cate gorical topology, and to those interested in categorical methods applied to the most common concrete categories. Categorical Closure Operators is self-contained and can be considered as a graduate level textbook for topics courses in algebra, topology or category theory. The reader is expected to have some basic knowledge of algebra, topology and category theory, however, all categorical concepts that are recurrent are included in Chapter 2. Moreover, Chapter 1 contains all the needed results about Galois connections, and Chapter 3 presents the the ory of factorization structures for sinks. These factorizations not only are essential for the theory developed in this book, but details about them can not be found anywhere else, since all the results about these factorizations are usually treated as the duals of the theory of factorization structures for sources. Here, those hard-to-find details are provided. Throughout the book I have kept the number of assumptions to a min imum, even though this implies that different chapters may use different hypotheses. Normally, the hypotheses in use are specified at the beginning of each chapter and they also apply to the exercise set of that chapter.
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.
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.
Inhaltsverzeichnis zu „Categorical Closure Operators “
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
Bibliographische Angaben
- Autor: Gabriele Castellini
- 2003, 2003, 300 Seiten, Maße: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer
- ISBN-10: 0817642501
- ISBN-13: 9780817642501
- Erscheinungsdatum: 15.05.2003
Sprache:
Englisch
Rezension zu „Categorical Closure Operators “
"The book...presents the contemporary theory of categorical closure operators, one of the main branches of categorical topology. The purpose of the theory is to develop a categorical characterization of the classical topological concepts. It provides a tool that allows us to extend these concepts to an arbitrary category so that the results of the theory have many applications in other fields of mathematics, particularly topology, algebra, and discrete mathematics....
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