Ebook: Categorical Closure Operators

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.

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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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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.
Content:
Front Matter....Pages i-xii
Front Matter....Pages 1-1
Galois Connections....Pages 3-8
Some Categorical Concepts....Pages 9-24
Factorization Structures For Sinks....Pages 25-40
Closure Operators: Definition and Examples....Pages 41-56
Idempotency, Weak Heredity and Factorization Structures....Pages 57-63
Additivity, Heredity, Suprema and Infima of Closure Operators....Pages 65-79
Additional Descriptions of ? and ? and Subobject Orthogonality....Pages 81-93
A Diagram of Galois Connections of Closure Operators....Pages 95-108
Regular Closure Operators....Pages 109-122
Hereditary Regular Closure Operators....Pages 123-127
Front Matter....Pages 129-129
Epimorphisms....Pages 131-135
Separation....Pages 137-163
Compactness....Pages 165-196
Connectedness....Pages 197-229
Connectedness in Categories with a Terminal Object....Pages 231-253
A Link between two Connectedness Notions....Pages 255-270
Different Constructions Related....Pages 271-277
Back Matter....Pages 279-300

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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.
Content:
Front Matter....Pages i-xii
Front Matter....Pages 1-1
Galois Connections....Pages 3-8
Some Categorical Concepts....Pages 9-24
Factorization Structures For Sinks....Pages 25-40
Closure Operators: Definition and Examples....Pages 41-56
Idempotency, Weak Heredity and Factorization Structures....Pages 57-63
Additivity, Heredity, Suprema and Infima of Closure Operators....Pages 65-79
Additional Descriptions of ? and ? and Subobject Orthogonality....Pages 81-93
A Diagram of Galois Connections of Closure Operators....Pages 95-108
Regular Closure Operators....Pages 109-122
Hereditary Regular Closure Operators....Pages 123-127
Front Matter....Pages 129-129
Epimorphisms....Pages 131-135
Separation....Pages 137-163
Compactness....Pages 165-196
Connectedness....Pages 197-229
Connectedness in Categories with a Terminal Object....Pages 231-253
A Link between two Connectedness Notions....Pages 255-270
Different Constructions Related....Pages 271-277
Back Matter....Pages 279-300
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