Simplicial Homotopy Theory
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Series Editors H. Bass
J. Oesterle A. Weinstein
Paul G. Goerss John F. Jardine Simplicial Homotopy Theory
Springer Basel AG
Authors: Paul G. Goerss Department of Mathematics University of Washington Seattle, WA 98195-4350 USA
John F. Jardine Department of Mathematics The University ofWestern Ontario London, Ontario N6A 5B7 Canada
e-mail: [email protected]
e-mail: [email protected]
1991 Mathematics Subject Classification 55-01; 18F25
Library of Congress Cataloging-in-Publication Data Goerss, Paul Gregory.
Simplicial homotopy theory / Paul G. Goerss, John F. Jardine. p. cm. (Progress in mathematics ; V. 174) Includes bibliographical references and index. ISBN 978-3-0348-9737-2 ISBN 978-3-0348-8707-6 (eBook) DOI 10.1007/978-3-0348-8707-6
1. Homotopy theory. 1. Jardine, J. F., 1951II. TitIe. III. Series: Progress in mathematics (Boston, Mass.) ; voI. 174. QA612.7.G64 1999 514'.24 -- dc21 Deutsche Bibliothek Cataloging-in-Publication Data Goerss, Paul G.:
Simplicial homotopy theory / Paul G. Goerss ; John F. Jardine. Basel ; Boston ; Berlin : Birkhăuser, 1999 (Progress in mathematics ; VoI. 174) ISBN 978-3-0348-9737-2
This work is subject to copyright. AlI rights are reserved, whether the whole or part of the material is concerned, specificalIy the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind ofuse whatsoever, permission from the copyright owner must be obtained. This book has been typeset with LAMSTeX © 1999 Springer Basel AG Originally published by Birkhiiuser Verlag in 1999 Printed on acid-free paper produced of chlorine-free pulp. TCF 00 ISBN 978-3-0348-9737-2
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PREFACE The origin of simplicial homotopy theory coincides with the beginning of algebraic topology almost a century ago. The thread of ideas started with the work of Poincare and continued to the middle part of the 20th century in the form of combinatorial topology. The modern period began with the introduction of the notion of complete semi-simplicial complex, or simplicial set, by EilenbergZilber in 1950, and evolved into a full blown homotopy theory in the work of Kan, beginning in the 1950s, and later Quillen in the 1960s. The theory has always been one of simplices and their incidence relations, along with methods for constructing maps and homotopies of maps within these constraints. As such, the methods and ideas are algebraic and combinatorial and, despite the deep connection with the homotopy theory of topological spaces, exist completely outside any topological context. This point of view was effectively introduced by Kan, and later encoded by Quillen in the notion of a closed model category. Simplicial homotopy theory, and more generally the homotopy theories associated to closed model categories, can then be interpreted as a purely algebraic enterprise, which has had substantial applications throughout homological algebra, algebraic geometry, number theory and algebraic K-theory. The point is that homo
