Simplicial Methods for Operads and Algebraic Geometry
This book is an introduction to two higher-categorical topics in algebraic topology and algebraic geometry relying on simplicial methods.Moerdijk’s lectures offer a detailed introduction to dendroidal sets, which were introduced by himself and Weiss as a
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Ieke Moerdijk • Bertrand Toën
Simplicial Methods for Operads and Algebraic Geometry Editors for this volume: Carles Casacuberta (Universitat de Barcelona) Joachim Kock (Universitat Autònoma de Barcelona)
Ieke Moerdijk Mathematisch Instituut Universiteit Utrecht Postbus 80.010 3508 TA Utrecht The Netherlands e-mail: [email protected]
Bertrand Toën I3M UMR 5149 Université Montpellier 2 Case Courrier 051 Place Eugène Bataillon 34095 Montpellier Cedex France e-mail: [email protected]
2010 Mathematics Subject Classification: primary: 55U40, 18G30; secondary: 55P48, 18D50, 18F10, 14A20 ISBN 978-3-0348-0051-8 DOI 10.1007/978-3-0348-0052-5
e-ISBN 978-3-0348-0052-5
© Springer Basel AG 2010 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. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
Foreword This book is an introduction to two higher-categorical topics in algebraic topology and algebraic geometry relying on simplicial methods. It is based on lectures delivered at the Centre de Recerca Matem` atica in February 2008, as part of a special year on Homotopy Theory and Higher Categories. Ieke Moerdijk’s lectures constitute an introduction to the theory of dendroidal sets, an extension of the theory of simplicial sets designed as a foundation for the homotopy theory of operads. The theory has many features analogous to the theory of simplicial sets, but it also reveals many new phenomena, thanks to the presence of automorphisms of trees. Dendroidal sets admit a closed symmetric monoidal structure related to the Boardman–Vogt tensor product. The lecture notes develop the theory very carefully, starting from scratch with the combinatorics of trees, and culminating with a model structure on the category of dendroidal sets for which the fibrant objects are the inner Kan dendroidal sets. The important concepts are illustrated with detailed examples. The lecture series by Bertrand To¨en is a concise introduction to derived algebraic geometry. While classical algebraic geometry studies functors from the category of commutative rings to the category of sets, derived algebraic geometry is concerned with functors from simplicial commutative rings (to allow derived tensor products instead of the usual ones) to simplicial sets (to allow derived quotients instead of the usual ones). The central objects are derived (higher) stacks, which are functors satisfying a certain up-to-homotopy descent condition. The lectures start with motivating examples from moduli theory, to move on to simplicial presheaves and algebraic (higher) stacks; next comes the homotopy theory of simplicial commutative rings, and finally everything comes
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