APPLIED CATEGORICAL STRUCTURES

A J oumal Devoted to Applications of Categorical Methods in Algebra, Analysis, Order, Topology and Computer Science

VOLUME 4-1996

KLUWER ACADEMIC PUBLISHERS

DORDRECHT/BOSTON/LONDON

EDITOR-IN-CHIEF:

R. LOWEN, Department of Mathematics and Computer Science, University of Antwerp, RUCA, Groenenborgerlaan 171, 2020 Antwerpen, Belgium (E-mail: lowen@ruca.ua.ac.be).

EDITORIAL BOARD:

1. Admnek, Technical University of

Braunschweig, Germany

B. Banaschewski, McMaster University, Canada

H. L. Bentley, University of Toledo, U.S.A. G. C. L. Brummer, University of Cape Town,

South Africa E. Colebunders, Vrije Universiteit Brussel,

Belgium P. J. Collins, University of Oxford, U.K. A. Cs~szar, E<>tv(js Lorand University,

Hungary H. Ehrig, Technical University Berlin,

Germany

M. Erne, Universiti1t Hannover, Germany

M. Fourman, University of Edinburgh, U.K. A. Fr5licher, Universit6 de Geneve,

Switzerland E. Giuli, Universita degli Studi di L'Aquila,

Italy J. Gray, University of Illinois at Urbana,

U.S.A. K. A. Hardie, University of Cape Town, South

Africa

H. Herrlich, Universiti1t Bremen, Germany

S. S. Hong, Sogang University, South Korea

M. Husek, Charles University, Czechia J. R. Isbell, State University of New York at

Buffalo, U.S.A. G. M. Kelly, University of Sydney, Australia D. C. Kent, Washington State University,

U.S.A. H. Kleisli, Universite de Fribourg, Switzerland J. D. Lawson, Louisiana State University,

U.S.A. F. E. J. Linton, Wesleyan University, U.S.A. R. Nakagawa, Ibaraki University, Japan L. D. Nel, Carleton University, Canada H.-E. Porst, Universiti1t Bremen, Germany

G. Preuss, Free University Berlin, Germany

D. Pumpltln, Fern Universiti1t Hagen, Germany H. R5hrl, University of California at San Diego,

U.S.A. S. Salbany, University of Zimbabwe,

Zimbabwe D. S. Scott, Carnegie Mellon University, U.S.A. Z. Semadeni, University of Warsaw, Poland M. B. Smyth, University of London, U.K. G. E. Strecker, Kansas State University, U.S.A. R. Street, Macquarie University, Australia W. Tholen, York University, Canada O. Wyler, Carnegie Mellon University, U.S.A.

ISSN 0927-2852

All rights reserved

ISBN-13: 978-94-010-6602-0 e-ISBN-13: 978-94-009-0263-3 001: 10.1007/978-94-009-0263-3

© 1996 Kluwer Academic Publishers

No part of the material protected by this copyright notice may be reproduced or utilised in any form or by any means, electronic or mechanical,

including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

APPLIED CATEGORICAL STRUCTURES

Vol.4 No.1 March 1996

SPECIAL ISSUE

CATEGORICAL TOPOLOGY

Guest Editor: Eraldo Giuli

Preface v

HORST HERRLICH / Compactness and the Axiom of Choice 1-14

MARIA MANUEL CLEMENTINO / On Categorical Notions of Compact Objects 15-29

G. RICHTER / Reflective Relatives of Adjunctions 31-41

H. L. BENTLEY, M. HUSEK, and R. G. ORI / The Katetov Dimension of Proximity Spaces 43-55

MIROSLA V HUSEK and ANNA TOZZI/Generalized Reflective cum Coreflective Classes in Top and Unif 57-68

E. LOWEN-COLEBUNDERS and G. SONCK / On the Largest Coreflective Cartesian Closed Subconstruct of Prtop 69-79

MICHAEL BARR and M. CRISTINA PEDICCHIO / Topological Spaces and Quasi -Varieties

LURDES SOUSA / a-Sober Spaces via the Orthogonal Closure Operator

MANUELA SOBRAL / Some Aspects of Topological Descent

DAVID HOLGATE / The Pullback Closure Operator and Generalisations of

81-85

87-95

97-106

Perfectness 107-120

TIIt! ADAMEK / A Remark on Fixed Points of Functors in Topological Categories 121-126

HANS-E. PORST / Concrete Categories Are Concretely Equivalent iff Their Uniquely Transportable Modifications Are Strict Concretely Iso-morphic 127-128

Scope. The primary goal of the journal is to promote communication and increase dissem­ination of new results and ideas among mathematicians and computer scientists who use categorical methods in their research.

The journal focuses on applications of results, techniques and ideas from Category The­ory to Mathematics, in particular Algebra, Analysis, Order and Topology and to Computer Science.

These include the study of specific topological, algebraic and algebraic-topological constructs, categorical investigations in functional analysis, in continuous order theory, in algebraic and logical type theory, in automata theory, in data bases and in languages. Furthermore the journal intends to follow the development of emerging fields in which the application of categorical methods proves to be relevant.

The journal will publish both carefully refereed research papers and survey papers. In all cases it strives for significance, originality, good exposition and the highest scientific quality in its publications.

These articles, and others, have also been published in the book: Categorical Topology by Eraldo Giuli, ISBN 978-94-010-6602-0

Photocopying. In the U.S.A.: This journal is registered at the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923. Authorisation to photocopy items for internal or personal use, or the internal or personal use of specific clients is granted by Kluwer Academic Publishers for users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service, provided that the fee of USD 8.50 per copy is paid directly to CCC. For those organisations that have been granted a photocopy licence by CCC, a separate system of payment has been arranged. The fee code for users of the Transactional Reporting Service is 0927-2852/96 USD 8.50. Authorisation does not extend to other kinds of copying, such as that for general distri­bution, for advertising or promotional purposes, for creating new collective works, or for resale. In the rest of the world: Permission to photocopy must be obtained from the copyright owner. Please apply to Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Applied Categorical Structures is published quarterly. 1996 Prices: Institutional: NLG 411,-/USD 263.00 including postage and handling. Indi­viduals: NLG 210,-/USD 124.00 including postage and handling. Published by Kluwer Academic Publishers, Spuiboulevard 50, P.O. Box 17, 3300 AA Dor­drecht, The Netherlands, and 101 Philip Drive, Norwell, MA 02061, U.s.A.

Printed on acid-free paper

Vaguely Defined Objects

Representations, Fuzzy Sets and Nonclassical Cardinality theory

by Mac:iej Wygralak, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznan, Poland

THEORY AND DECISION LIBRARY 8: Mathematical and Statistical Methods 33

This unique monograph explores the cardinal, or quantitative, aspects of objects in the presence 01 vagueness, called vaguely defined objects. In the first part of the book such topics as fuzzy sets and

derivative ideas, twofold fuzzy sets. and flow sets are concisely reviewed as typical mathematical representations of vaguely delined objects. Also, a unifying, approximative representation is presented. The second part uses this representation, together with

&lslrok;ukasiewicl logic as a basis lor constructing a complete, general and easily applicable nonclassical cardinality theory for vaguely defined objects. Applications to computer and infOfIDation science are discussed. Audience: This volume will be 01 interest to mathematicians.

computer and information scientists. whose work involves mathematical aspects 01 vagueness. fuzzy sets and their methods. applied many-valued logics. e~pert systems and data bases.

Contents: Preface. Part I: Vaguely Defined Objects. 1. Basic Notions and Problems. 2. Mathematical Approaches to Vaguely Defined Objects. 3. Mathematical Approaches to Subdefinite Sets. 4. A Unifying ApproximatIVe Approach to Vaguely Defined Objects. Part II : Nonclassical Cardinality Theory for Vaguely Defined Objects. 5. EqUipotencles. 6. Generalized Cardinal Numbers. 7. Selected Applications. 8. Inequalities. 9. Many-Valued Generalizations. 10. Towards Arithmetical Operations. ". Addition. 12. Multiplication. 13. Other Basic Operations. 14. GeneraliZed Arithmetical Operations. 15. Cardinalities with Free Represenling Pairs. 16. Further Modifications and Final Remarks. Footnotes, Comments and Bibliographical Remarks. Bibliography. tndex 01 Definitions and Theorems. Ust 01 Symbots. Inde~ .

t996. 288 pp. ISBN 0·7923-3850-2 Hardbound NLG 210.00 I USD 147.00 I GBP 95.00

P.O. Box 322, 3300 AH Dordrecht, The Netherlands P.O. Box 358, Accord Station, Hingham,

MA

Discrete Analysis and Operations Research

edited by Alekse D. Korshunov, Sobolov Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

MATHEMATICS AND ITS APPLICATIONS 355

The contributions to this volume have all been translated from the fi rst volume of the Russian journal Discrete Analysis and Operational Research, published at the Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia, in 1994. The papers collected here give an excellent

overview of recent Russian research in topics such as analysis of algorithms, combinatorics, graphs, lower bounds for complexity of Boolean functions, packing and coverings, scheduling theory, search and sorting, linear programming, and testing. Audience: This book will be of interest to

specialists in discrete mathematics and computer science, and engineers.

1996, 352 pp. ISBN 0-7923-3866-9 Hardbound NLG 265.00 I USD 185.00 I GBP 119.00

P.O. Box 322, 3300 AH Dordrecht. The Netherlands P.O. Box 358, Accord Station, Hingham,

~1A 02018·0358,U.S.A.

Preface

This issue contains a selected group Qf papers presented during the International Workshop on Categorical Topology held from August 31 - September 4, 1994 at the University of L' Aquila, L' Aquila, Italy.

The workshop was organized by the Department of Pure and Applied Mathe­matics of the University of L' Aquila and was made possible by grants from the Department, the University, the Regione Abruzzo and the Cassa Di Risparmio della Provincia de l' Aquila. We wish to thank the officers of the University and public local administrations who facilitated the financial supports.

We would also like to thank the INFN (Istituto Nazionale di Fisica Nucleare) for giving the opportunity to all participants to visit the impressive laboratories under Gran Sasso and the major of Roseto who organized a nice party in the garden of the Villa Comunale.

The meeting was organized by E. Giuli, H. Herrlich and A. Tozzi and was attended by 45 mathematicians from 11 countries.

The program consisted of 34 talks on various aspects of categorical topology. We would like to express our gratitude to the Dean of the Faculty of Science

Umberto Villante for the friendly welcome he extended to the participants on behalf of the University of L' Aquila and to the Head of the Department Josef Myjak for his encouragement.

Thanks are due to the many referees of the papers presented for publication in this issue for their indispensable assistance in selecting and improving the papers.

Finally, we would like to thank the staff ofthe "Aula Magna" who were always ready to give announcements on the big screen, for the invaluable technical assis­tance, and the students S. Chiappini, M. Giuli, C. Lattanzio, G. Ippoliti and E. Lucrezi, for their help in the organization.

ERALDO GIULI