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Categorical Closure Operators

  • Textbook
  • © 2003

Overview

Authors:
  1. Gabriele Castellini

    1. Department of Mathematics Mayaguez, University of Puerto Rico, Puerto Rico, USA

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  • Castellini has worked in the area of categorical closure operators for over 15 years
  • Combines theory with practice
  • Includes many helpful examples to illustrate the concepts discussed
  • Provides exercises for those interested in self-study
  • Includes applications to classical concepts

Part of the book series: Mathematics: Theory & Applications (MTA)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xii
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  2. General Theory

    1. Front Matter

      Pages 1-1
      Download chapter PDF

    2. Galois Connections

      • Gabriele Castellini
      Pages 3-8

    3. Some Categorical Concepts

      • Gabriele Castellini
      Pages 9-24

    4. Factorization Structures For Sinks

      • Gabriele Castellini
      Pages 25-40

    5. Closure Operators: Definition and Examples

      • Gabriele Castellini
      Pages 41-56

    6. Idempotency, Weak Heredity and Factorization Structures

      • Gabriele Castellini
      Pages 57-63

    7. Additivity, Heredity, Suprema and Infima of Closure Operators

      • Gabriele Castellini
      Pages 65-79

    8. Additional Descriptions of Ĉ and Č and Subobject Orthogonality

      • Gabriele Castellini
      Pages 81-93

    9. A Diagram of Galois Connections of Closure Operators

      • Gabriele Castellini
      Pages 95-108

    10. Regular Closure Operators

      • Gabriele Castellini
      Pages 109-122

    11. Hereditary Regular Closure Operators

      • Gabriele Castellini
      Pages 123-127

  3. Applications

    1. Front Matter

      Pages 129-129
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    2. Epimorphisms

      • Gabriele Castellini
      Pages 131-135

    3. Separation

      • Gabriele Castellini
      Pages 137-163

    4. Compactness

      • Gabriele Castellini
      Pages 165-196

    5. Connectedness

      • Gabriele Castellini
      Pages 197-229

    6. Connectedness in Categories with a Terminal Object

      • Gabriele Castellini
      Pages 231-253

    7. A Link between two Connectedness Notions

      • Gabriele Castellini
      Pages 255-270

    8. Different Constructions Related

      • Gabriele Castellini
      Pages 271-277
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Keywords

About this book

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.

Reviews

"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....

The benefit of the book is that, after introducing the reader to the general theory, it presents a number of applications based on quite recent results achieved in this field. Thus the book provides basic literature for those interested in the theory of closure operators and its applications. It is an excellent exposition of a part of modern mathematics."

— INTERNATIONALE MATHEMATISCHE NACHRICHTEN

 

"The book under review puts its emphasis on giving a fairly elementary introduction in to the basic categorical theory of closure operators (Part I) and of their use in categorical topology (Part II), paying particular attention to the author's work on notions of connectedness...

Each of the 17 section sof the book ends with a list of references for suggested further reading and a set of exercises, many of which ask the reader to verify claims made earlier in the text." ---ZENTRALBLATT MATH

 

Authors and Affiliations

  • Department of Mathematics Mayaguez, University of Puerto Rico, Puerto Rico, USA

    Gabriele Castellini

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