Table Of ContentTrends in Mathematics
Series Titles
AndersonlLapidus (eds). Progress in Inverse Spectral Geometry
JainlRizvi (eds). Advances in Ring Theory
Martino/Kantor/LunardonlPasiniiTamburini (eds). Groups and Geometries
Ross/AndersonlLitvinov/SinghlSunderlWildberger (eds). Harmonic Analysis
and Hypergroups
Karatzas/RajpuUTaqqu (eds). Stochastic Processes and Related Topics:
In Memory of Stamatis Cambanis 1943-1995
KomatsulKuranishi (eds). Analysis and Geometry in Several
Complex Variables: Proceedings of the 40th Taniguchi Symposium
BirgeUMargolis/MeakinlSapir (eds). Algorithmic Problems in
Groups and Semigroups
EllingsrudlFultonlVistoli (eds). Recent Progress in Intersection Theory
Passi (ed). Algebra: Some Recent Advances
Eklof/Gobel (eds). Abelian Groups and Modules
Gyr/KinzelbachlTsinober (eds). Fundamental Problematic Issues in Turbulence
BambahlDumirlHans-Gill (eds). Number Theory
Rebolledo (ed). Stochastic Analysis and Mathematical Physics:
ANESTOC '98, Proceedings of the Third International Workshop
HidalKarandikar/KunitalRajpuUWatanabe/Xiong (eds). Stochastics in Finite
and Infinite Dimensions: In Honor of Gopinath Kallianpur
Birkenmeier/ParklPark (eds). International Symposium on Ring Theory
Koslowski/Melton (eds). Categorical Perspectives
Trends in Mathematics
Trends in Mathematics is a book series devoted to focused collections of articles arising
from conferences, workshops or series of lectures.
Topics in a volume may concentrate on a particular area of mathematics, or may
encompass a broad range of related subject matter. The purpose of this series is both
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Ideally, volumes should not exceed 350-400 pages in length.
Proposals to the Publisher are welcomed at either:
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Categorical Perspectives
Jurgen Koslowski
Austin Melton
Editors
Springer Science+Business Media, LLC
Jlirgen Koslowski Austin Melton
Institut fiir Theoretische Informatik Department of Mathematics
TU Braunschweig and Computer Science
38023 Braunschweig Kent State University
Germany Kent, OH 44242
U.S.A.
Library of Congress Cataloging-in-Publication Data
Categorical perspectives / Jiirgen Koslowski, Austin Melton, editors.
p. cm.-(Trends in mathematics)
Includes bibliographical references.
ISBN 978-1-4612-7117-8 ISBN 978-1-4612-1370-3 (eBook)
DOI 10.1007/978-1-4612-1370-3
1. Categories (Mathematics) 1. Koslowski, Jiirgen, 1956- II. Melton, Austin. III.
Series.
QA169.C35 2001
511.3-dc21 2001025482
CIP
AMS Subject Classifications: 03E25, 05CIO, 05C76, 06A 15, 18-01, 18-02, 18A02, 18A05, 18AIO,
18A15, 18A20, 18A22, 18A30, 18A32, 18A35, 18A40, 18B05, 18D15, 54A05, 54A20, 54C35,
54D20, 54030, 54E05, 54E15
Printed on acid-free paper
© 2001 Springer Science+Business Media New York
Originally published by Birkhauser Boston in 2001
Softcover reprint of the hardcover 1s t edition 2001
AII 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 rreely
byanyone.
ISBN978-1-4612-7117-8 SPIN 10761690
Typeset by the editors in LATEX
987 6 5 4 3 2 1
Dedicated to George E. Strecker
on the occasion of his sixtieth birthday
Contents
Preface
Jiirgen Koslowski and Austin Melton ................................. vii
Categories: A Free Tour
Lutz Schroder ......................................................... 1
The Functor that Wouldn't be
Y. T. Rhineghost ..................................................... 29
The Emergence of Functors
Y. T. Rhineghost ..................................................... 37
Too Many Functors
Alois Zmrzlina ....................................................... 47
Contributions and Importance of
Professor George E. Strecker's Research
Jiirgen Koslowski ..................................................... 63
10 Rules for Surviving as a Mathematician and Teacher
George E. Strecker ................................................... 91
Connections and Polarities
Austin Melton ........................................................ 95
Categorical Closure Operators
G. Castellini ........................................................ 109
Extensions of Maps from Dense Subspaces
H. L. Bentley ....................................................... 151
Characterizations of Subspaces of Important Types of
Convergences Spaces in the Realm of Convenient Topology
Gerhard PreufJ ...................................................... 177
Vlll Contents
The Naturals are Lindelof iff Ascoli Holds
Y. T. Rhineghost .................................................... 191
Revisiting the Celebrated Thesis of J. de Groot:
"Everything is Linear."
Ludvik Janos ........................................................ 197
Finite Ultrametric Spaces and Computer Science
Vladimir A. Lemin .................................................. 219
The Copnumber of a Graph is Bounded by l ~ genus (G) J + 3
Bernd S. W. Schroeder .............................................. 243
Abelian Groups: Simultaneously Reflective and
Coreflective Subcategories versus Modules
Robert El Bashir, Horst Herrlich and Miroslav Husek ................ 265
Preface
This book had its beginnings in a conference held in honor of Professor
George E. Strecker's 60th birthday. The conference was held in August
1998 on the campus of Kent State University.
During the planning for the conference, questions about a conference pro
ceedings came up. It was not, however, until discussions with Birkhauser
started that the final format for this book began to take shape. Bernd
Schroeder introduced us to Joseph Kung who helped us prepare materi
als for Birkhauser and introduced us to Ann Kostant, Executive Editor
for Mathematics and Physics at Birkhauser. She and Tom Grasso, also
of Birkhauser, encouraged us to have expository and research chapters,
and in discussions with prospective authors, the idea of including teaching
chapters emerged. The end result is a delightful combination of teaching,
expository, and research papers.
One of the teaching chapters, 10 Rules for Surviving as a Mathematician
and Teacher presents in a humorous and insightful manner suggestions
for being a successful mathematician and a successful teacher. The other
teaching chapters, written by Alois Zmrzlina and Y.T. Rhineghost, are
tutorials in category theory; these tutorials are presented as dialogs between
master teachers and their students.
Some of the chapters are mainly tutorial in nature, and some are mainly
research. Of course, many are a combination of the two. Since a major
emphasis of Professor Strecker's research has been category theory and its
uses to explain and clarify other areas, mainly topology, it is natural that
many chapters in this book use category theory in a similar manner - thus,
the title Categorical Perspectives.
An introduction to category theory in general, and to this book in partic
ular, is the chapter "Categories: A Free Tour" by Lutz Schroder. The chap
ter "Contributions and Importance of Professor George Strecker's
Research" by Jiirgen Koslowski is also a good introduction to this book
and to the more specialized field of categorical topology.
x Preface
We would like to extend a special thank you to Joyce Fuell, who was
extremely helpful in formatting and preparing this book for publication.
Jiirgen Koslowski
Institut fUr Theoretische Informatik
Technische Universitaat Braunschweig
Braunschweig, Germany
[email protected]
Austin Melton
Department of Mathematics and Computer Science
Kent State University
Kent, OR 44242 (USA)
[email protected]
