Lecture Notes in Mathematics
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1 Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann 1348 F. Borceux (Ed.) Categorical Algebra and its Applications Proceedings of a Conference, held in Louvain-La-Neuve, Belgium, July 26 - August 1, 1987 Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
2 Editor Francis Borceux Departement de Mathematiques, Universite Catholique de Louvain 2 chemin du Cyclotron, 1348 Louvain-La-Neuve, Belgium Mathematics Subject Classification (1980): ISBN Springer-Verlag Berlin Heidelberg New York ISBN Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hernsbach/Berqstr. 2146/
3 INTRODUCTION The first Louvain-la-Neuve Conference on Categorical Algebra and its Applications has been held in Louvain-la-Neuve (Belgium) from July 26 to August 1, The organizing committee was composed of: J. Adamek (Prague, Czecoslovakia) A. Kock (Aarhus, Denmark) F.W. Lawvere (Buffalo, USA) R. Walters (Sydney, Australia). The local organizers were: F. Borceux (Louvain-la-Neuve, Belgium) F. Waninier (Leuven, Belgium) 128 persons did take part in the Conference and 96 of them did deliver a talk. The organizers would like to express their gratitude to the various organisms who did support financially the meeting and therefore made it accessible to a very large international audience: Le Fonds National de la Recherche Scientifique L'Universite Catholique de Louvain De Katholieke Universiteit te Leuven Le Centre Belge de Recherches en Mathematiques Le Centre National de Recherches en Logique Apple Computer Les commercants de Louvain-la-Neuve. The proceedings volume has been realized with the competent collaborationof 46 referees who did study carefully as many papers; the editor expresses his gratitude to all of them. The material submitted for these proceedings represented not far from 800 pages. Since just halfof this material could be published, some very painful choices had to be made and it is certainly a pity that some good papers had to be rejected; the editor hopes they will find a good home somewhere else.
4 TABLE OF CONTENTS J. Adamek, J. Rosicky, V. 'Irnkova; Are all limit-closed subcategories of locally presentable categories reflective?. 1 M. Barr; On categories with effective unions. 19 F. Borceux t, J. Pelletier; Descent theory for Banach modules. 36 D. Bourn; Pseudofunctors and non-abelian weak equivalences. 55 A.M. Cegarra, A.R. Garzon, P. Carrasco; An exact sequence in the first variable for non-abelian cohomology in algebraic categories. A Mayer- Vietoris sequence for non-abelian cohomology of groups. 72 Y. Diers; Locally Hilbert categories. 87 F. Druck, G. Reyes; Sur la convergence de suites de fonctions suivant des filtres. 102 J. Duskin; The Azumaya complex of a commutative ring. 107 L. Espaiiol; The spectrum lattice of Baer rings and polynomials. 118 F. Gago; Morse germs in S. D. G S. Ghilardi, G.C. Meloni; Modal and tense predicate logic: models in presheaves and catego rical conceptualization H. Herrlich; On the representability of partial morphisms in I2.1!. and in related constructs J. Isbell, 1. KHz, A. Pultr, J. Rosicky; Remarks on localic groups. 154 P.T. Johnstone, Sun Shu-Hao; Weak products and Hausdorff locales. 173 A. Kock; Generalized fibre bundles. J. Koslowski; Closure operators with prescribed properties. J. Lambek; On the unity of algebra and logic. A. Lirola, E.R. Aznar, M. Bullejos; The cohomology groups of an epimorphism. 230 S. Mac Lane; Decker's sharper Kunneth formula. M. Mawanda; On a categorical analysis of Zadeh generalized subsets of sets I. G.C. Meloni, E. Rogora; Global and infinitesimal obeervablee. 1. Moerdijk; Toposes and groupoids. S. Niefield, K. Rosenthal; Componental nuclei. G. Rosolini; Representation theorems for p-categories. z. Semadeni; On the category of compact convex sets and related categories. R. Street; Fillers for nerves. V. Trnkova: Simultaneous representations in categories. W. Weiss; Uniform spaces can be represented by completely distributive lattices t This paper has been edited by A. Kock
5 LIST OF PARTICIPANTS J. Adam.ek; Prague, Csecoslovakia, I. Aitchison; Melbourne, Australia. C. Anghel; Dingolfing, Germany. E. Aznar; Granada, Spain. B. Banaschewski; Hamilton, Canada. L. Barbieri; Genova, Italy. J. Barja; Malaga, Spain. M. Barr; Montreal, Canada. J. Benabou; Paris, France. H. Bentley; Toledo, USA. F. Borceux; Lcuvain-la-Neuve, Belgium. R. Borger; Hagen, Germany. D. Bourn; Amiens, France. D. Brandt; Garbsen,Germany. s. Breitsprecher; Tiibingen, Germany. H. Brinkmann; Konstanz, Germany. M. Bullejos; Granada, Spain. P. Carrasco; Granada, Spain. G. Castelliw; Mayaguez, Puerto Rico. P. Cherenack; Cape Town, South Africa. E. Colebunders; Brussel, Belgium. R. Cruciani; Roma, Italy. V. De Paiva; Cambridge, United Kingdom. F. De Vries; Utrecht, Netherlands. Y. Diers; Valenciennes, France. E. Dubuc; Buenos Ayres, Argentine. P. Dupont; Louvein-la-Neuve, Belgium. S. Eilenberg; New York, USA. L. Espafiol] Logrono, Spain. A. Frei; Vancouver, Canada. A. FrOlicher; Geneve, Switzerland. F. Gago; Santiago de Compostela, Spain. s. Ghilardi; Bergamo, Italy. M. Grandis; Genova, Italy. J. Gray; Urbana, USA. R. Guitart; Paris, France. H. Herrlich; Bremen, Germany. P. Higgins; Durham, United Kingdom; R. Hoffmann; Bremen, Germany. H. Husekj Prague, Csecoslovakia. J. Isbell; Buffalo, USA. G. Jarzembski; Torun, Poland. J. Johnson; New Brunswick, USA. M. Johnson; Sydney, Australia. P. Johnstone; Cambridge, Un. Kingdom. A. Joyal; Montreal, Canada. K. Kam.ps; Hagen, Germany. s. Kasangian; Milano, Italy. G. Kelly; Sydney, Australia. H. Kleisli; Fribourg, Switzerland, A. Kock; Aarhus, Denmark. M. Korostenski; Johannesburg, S. Africa. J. Koslowski; Nashville, USA. I. KHz; Prague, Csecoslovakia, A. Labella; Roma, Italy. J. Lambek; Montreal, Canada. R. Lavendhomme; Louvain-la-Nv, Belgium. F. Lawvere; Buffalo, USA. P. Lecouturier; Kinshasa, Zai.'re. D. Lever; Nova Scotia, Canada. F. Linton; Middletown, USA. A. Lirola Terrez; Almeira, Spain. B. Loiseau; Louvain-la-Neuve, Belgium. R. Lowen; Antwerpen, Belgium. L. Lubikulu; Louvain-la-Neuve, Belgium. T. Lucas; Louvain-la-Neuve, Belgium. J. Mac Donald; Vancouver, Canada. K. Mackenzie; Durham, United Kingdom. s. Mac Lane; Chicago, USA. R. Macleod; Slickville, Canada. S. Mantovani, Parma, Italy. S. Mawanda; Louvain-le-Neuve, Belgium. G. Melonij Milano, Italy. M. Minguez; Logrono, Spain. I. Moerdijk; Amsterdam, Netherlands. T. Mubanga; Louvain-Is-Neuve, Belgium. J. Navaro; Badajos, Spain.
6 VIII L. Nel; Ottawa, Canada. S. Niefield; Schenectady, USA. A. Obtulowicz; Warsawa, Poland. D. Pape; Berlin, Germany. R. Pare; Halifax, Canada. D. Pavlovic; Utrecht, Netherlands. M. Pedicchio; Trieste, Italy. J. Pelletier; Toronto, Canada. M. Pfender; Berlin, Germany. A. Pitts; Brighton, United Kingdom. T. Porter; Bangor, United Kingdom. J. Power; Sydney, Australia. J. Pradines; Toulouse, France. A. Pultr; Prague, Czecoslovakia. D. Piimplun; Hagen, Germany. J. Reiterman; Prague, Czecoslovakia. G. Reyes; Montreal, Canada. J. Riguet; Paris, France. E. Robinson; Cambridge, Un. Kingdom. E. Rogora; Milano, Italy. J. Roisin; Louvain-la-Neuve, Belgium. R. Rosebrugh; Sackville, Canada. K. Rosenthal; Schenectady, USA. J. Rosieky; Brno, Csecoslovalia. G. Rosolini; Parma, Italy. F. Rossi; Trieste, Italy. S. Salbany; Harare, Zimbabwe. J. Sancho de Salas; Badajos, Spain. M. Sancho de Salas; Badajos, Spain. L. Schinaia; Roma, Italy. D. Schumacher; Wolfville, Canada. F. Schwarz; Toledo, USA. S. Semadeni; Warsawa, Poland. H. Simmons; Aberdeen, United Kingdom. R. Street; Sydney, Australia. P. Taylor; London, United Kingdom. M. Thiebaud; Thonex, Switzerland. W. Tholen; Toronto, Canada. M. Tierney; New Brunswick, USA. V. Topencharov; Sofia, Bulgaria. V. TrnkOV8.; Prague, Czecoslovakia. J. Turpin; Louvain-la-Neuve, Belgium. K. Ulbich; Hamburg, Germany. G. Van den bossche; Louvain-le-N,, Belgium. B. Veit; Roma, Italy. J. Vermeulen; Pretoria, South Africa. R. Walters; Sydney, Australia. F. Warrinier; Leuven, Belgium. S. Week; Toledo, USA. W. Weiss; Darmstadt, Germany.
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