Categorical Closure Operators

299 Pages·2003·13.865 MB·English

by  Gabriele Castellini (auth.)

Mathematics: Theory & Applications
Series Editor
NolanWallach
Gabriele Castellini
Categorical Closure
Operators
Springer Science+Business Media, LLC
Gabriele Castellini
University of Puerto Rieo
Department of Mathematics
Mayaguez,00681-9018
Puerto Rieo
U.S.A.
Library of Congress Cataloging-in-Publication Data
Castellini, Gabriele.
Categorical closure operators / Gabriele Castellini.
p. cm.-(Mathernatics : theory & applications)
Includes bibliographical references and index.
ISBN 978-1-4612-6504-7 ISBN 978-0-8176-8234-7 (eBook)
DOI 10.1007/978-0-8176-8234-7
1. Categories (Mathernatics) 2. Closure operators. 1. Title. 11. Series.
QA169.C342003
511.3-dc21
AMS Subject Classifications: 18-01, 06A15
Printed on acid-free paper.
©2003 Springer Science+ Business Media N ew York
Originally published by Birkhäuser Boston in 2003
Softcover reprint of the hardcover 1s t edition 2003
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher Springer Science+Business Media, LLC
except for brief excerpts in connection with reviews or
scholarly analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereafter
developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by the
Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-6504-7 SPIN 10841238
Typeset by the author in U\TEX.
9 8 7 6 5 4 3 2 1
To my parents) my wife and son
Contents
Preface ix
Introduction xi
I GENERAL THEORY 1
1 Galois Connections 3
2 Some Categorical Concepts 9
3 Factorization Structures For Sinks 25
4 Closure Operators:Definition and Examples 41
5 Idempotency, Weak Heredity and Factorization
Structures 57
6 Additivity, Heredity,Suprema and Infima of
Closure Operators 65
7 Additional Descriptions of Cand Cand Subobject
Orthogonality 81
Contents
Vlll
8 A Diagram of Galois Connections of Closure Operators 95
9 Regular Closure Operators 109
10 Hereditary Regular Closure Operators 123
II APPLICATIONS 129
11 Epimorphisms 131
12 Separation 137
13 Compactness 165
14 Connectedness 197
15 Connectedness in Categories with a Terminal Object 231
16 A Link between two Connectedness Notions 255
17 Different Constructions Related 271
References 279
List of Symbols 289
Index 295
Preface
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
agraduateleveltextbookfortopicscoursesin algebra, topologyor 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
essentialfor 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 ofthe theory offactorization 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.
When no assumptions are specified, the ones from the previous chapter
are still in use. However, a general guide for the reader is the following.
Throughout the book the standard setting is a category X with pullbacks
andarbitraryintersections, which isalsoassumedto be an (E,M)-category
x Preface
for sinks. These will be referred to as the basic assumptions since they will
be used in all chapters of the book. In Chapters 9-11, the existence of
equalizers and the fact that M contains all regular monomorphisms are
added. In Chapter 12, the existence of finite products is added. However,
Chapter 13 goes back to the basic assumptions with, in addition,just the
existence of finite products, even though later in the chapter the assump
tion of E consisting of episinks is made. In Chapters 14-17, again just the
basic assumptions are used and the existence ofa terminal object is added
for Chapter 15only. Of course, it should be understood that when a result
needs a particularassumption, it isincluded in thestatement ofthat result.
Everychapter ends with a number ofexercises that either ask the reader
to verify a claim in a concrete situation, to fill in some missing details in
a proof, or even to try to analyze some situations that in the book are
not specifically dealt with. They should provide enough challenge to those
readers who want to test their understanding of the material.
The bookincludes afair numberofexamplesin the categoriesoftopolog
ical spaces, fuzzy topological spaces, groups and abelian groups. Of course
there are some occasionalexamples that do not belong to any ofthe above.
Details about examples are hardly ever given.This has a double purpose in
that, while keeping the book smaller, it provides the instructor with plenty
of exercise material if the book is used as a textbook in a course.
Many references have been included at the end of each chapter for the
reader who wants to consult the original works.
Finally, Paul Taylor's Commutative Diagramsin 'lEX macro packagehas
proved an essential tool for typesetting nearly all the included diagrams.
G. Castellini
Department of Mathematics
University of Puerto Rico
Mayagiiez Campus
Introduction
Undoubtedly, the inspiringworkforthe theoryofcategoricalclosure opera
tors wasSalbany'spaper [S]. In this papera particularclosureconstruction
in the category Top of topological spaces was introduced. This construc
tion wasthen extended by other authors to an arbitrary category X and in
an indirect wayit ledto the general concept ofcategoricalclosureoperator.
The first one to see in Salbany's closure construction a great potential for
further development wasEraldo Giuli whoin [Gl] used it to obtain a char
acterizationofthe epimorphisms inepireflectivesubcategories ofTop. The
first paper to present a more formal introduction of the above operator in
Top was[DGIJ.This wasfollowedby [DG2] and [GHIJin which a diagonal
theorem for quotient reflective subcategories ofTop wasproved and some
questions about co-well-poweredness ofepireflective subcategories of Top
wereanswered.
The first attempt to introduce a general notion of closure operator in
a concrete category was made by Castellini in [Cl]. In this paper an ex
tended version ofSalbany's construction wasused to study the surjectivity
of epimorphisms in several subcategories of abelian groups. Moreover, a
dual notion was used to study the monomorphisms. Finally in [DG3] the
current notion ofcategorical closure operator wasintroduced in a category
X together with some important basic properties. This paper laid the ba
sis for further development of the theory. For instance [CS], [GMT] and
[TIJdealt with the diagonal theorem in an arbitrary category, among other
topics. In [KIJ some of the results in [DG3] were sharpened. Among the
other papers that have dealt with the general theory ofcategorical closure