Simplicial Methods for Operads and Algebraic Geometry

  • Ieke Moerdijk
  • Bertrand Toën

Part of the Advanced Courses in Mathematics - CRM Barcelona book series (ACMBIRK)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Lectures on Dendroidal Sets

    1. Front Matter
      Pages 1-3
    2. Ieke Moerdijk, Bertrand Toën
      Pages 5-10
    3. Ieke Moerdijk, Bertrand Toën
      Pages 11-21
    4. Ieke Moerdijk, Bertrand Toën
      Pages 23-39
    5. Ieke Moerdijk, Bertrand Toën
      Pages 41-54
    6. Ieke Moerdijk, Bertrand Toën
      Pages 55-67
    7. Ieke Moerdijk, Bertrand Toën
      Pages 69-77
    8. Ieke Moerdijk, Bertrand Toën
      Pages 79-92
    9. Ieke Moerdijk, Bertrand Toën
      Pages 93-115
  3. Simplicial Presheaves and Derived Algebraic Geometry

    1. Front Matter
      Pages 119-119
    2. Ieke Moerdijk, Bertrand Toën
      Pages 121-126
    3. Ieke Moerdijk, Bertrand Toën
      Pages 127-141
    4. Ieke Moerdijk, Bertrand Toën
      Pages 143-158
    5. Ieke Moerdijk, Bertrand Toën
      Pages 159-165
    6. Ieke Moerdijk, Bertrand Toën
      Pages 167-177
    7. Ieke Moerdijk, Bertrand Toën
      Pages 179-184
  4. Back Matter
    Pages 185-186

About this book

Introduction

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 foundation for the homotopy theory of operads. The theory of dendroidal sets is based on trees instead of linear orders and has many features analogous to the theory of simplicial sets, but it also reveals new phenomena. For example, dendroidal sets admit a closed symmetric monoidal structure related to the Boardman–Vogt tensor product of operads. The lecture notes start with the combinatorics of trees and culminate with a suitable model structure on the category of dendroidal sets. Important concepts are illustrated with pictures and examples.

The lecture series by Toën presents 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) to simplicial sets (to allow derived quotients). The central objects are derived (higher) stacks, which are functors satisfying a certain up-to-homotopy descent condition. These lectures provide a concise and focused introduction to this vast subject, glossing over many of the technicalities that make the subject’s research literature so overwhelming.

Both sets of lectures assume a working knowledge of model categories in the sense of Quillen. For Toën’s lectures, some  background in algebraic geometry is also necessary.

Keywords

Grad Homotopy algebraic geometry commutative algebra commutative ring homotopy theory

Authors and affiliations

  • Ieke Moerdijk
    • 1
  • Bertrand Toën
    • 2
  1. 1.Mathematisch InstituutRijksuniversiteit UtrechtUtrechtNetherlands
  2. 2.UMR CNRS 5580, Labo. Mathématiques Emile PicardUniversité Toulouse IIIToulouse CX 4France

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-0052-5
  • Copyright Information Springer Basel AG 2010
  • Publisher Name Springer, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0348-0051-8
  • Online ISBN 978-3-0348-0052-5
  • About this book