Categories and Sheaves
Categories and sheaves, which emerged in the middle of the last century as an enrichment for the concepts of sets and functions, appear almost everywhere in mathematics nowadays. This book covers categories, homological algebra and sheaves in a systematic
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Series editors M. Berger B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin L. Hörmander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane B. Totaro A. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates
S.R.S. Varadhan
332
Masaki Kashiwara Pierre Schapira
Categories and Sheaves
ABC
Masaki Kashiwara
Pierre Schapira
Research Institute for Mathematical Sciences Kyoto University Kitashirakawa-Oiwake-cho 606-8502 Kyoto Japan E-mail: [email protected]
Institut de Mathématiques Université Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05, France E-mail: [email protected]
Library of Congress Control Number: 2005930329 Mathematics Subject Classification (2000): 18A, 18E, 18F10, 18F20, 18G ISSN 0072-7830 ISBN-10 3-540-27949-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-27949-5 Springer Berlin Heidelberg New York 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
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Preface
The language of Mathematics has changed drastically since the middle of the twentieth century, in particular after Grothendieck’s ideas spread from algebraic geometry to many other subjects. As an enrichment for the notions of sets and functions, categories and sheaves are new tools which appear almost everywhere nowadays, sometimes simply in the role of a useful language, but often as the natural approach to a deeper understanding of mathematics. Category theory, initiated by Eilenberg and Mac Lane in the forties (see [19, 20]), may be seen as part of a wider movement transcending mathematics, of which structuralism in various areas of knowledge is perhaps another facet. Before the advent of categories, people were used to working with a given set endowed with a given structure (a topological space for example) and to studying its properties. The categorical point of view is essentially different. The stress is placed not upon the objects, but on the rel
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