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  • © 2010

Simplicial Methods for Operads and Algebraic Geometry

Birkhäuser

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  • The exposition has maintained the directness of the original lectures

  • Includes supplementary material: sn.pub/extras

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

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-ix
  2. Lectures on Dendroidal Sets

    1. Front Matter

      Pages 1-3
    2. Operads

      • Ieke Moerdijk, Bertrand Toën
      Pages 5-10
    3. Trees as operads

      • Ieke Moerdijk, Bertrand Toën
      Pages 11-21
    4. Dendroidal sets

      • Ieke Moerdijk, Bertrand Toën
      Pages 23-39
    5. Tensor product of dendroidal sets

      • Ieke Moerdijk, Bertrand Toën
      Pages 41-54
    6. A Reedy model structure on dendroidal spaces

      • Ieke Moerdijk, Bertrand Toën
      Pages 55-67
    7. Boardman–Vogt resolution and homotopy coherent nerve

      • Ieke Moerdijk, Bertrand Toën
      Pages 69-77
    8. Inner Kan complexes and normal dendroidal sets

      • Ieke Moerdijk, Bertrand Toën
      Pages 79-92
    9. Model structures on dendroidal sets

      • Ieke Moerdijk, Bertrand Toën
      Pages 93-115
  3. Simplicial Presheaves and Derived Algebraic Geometry

    1. Front Matter

      Pages 119-119
    2. Motivation and objectives

      • Ieke Moerdijk, Bertrand Toën
      Pages 121-126
    3. Simplicial presheaves as stacks

      • Ieke Moerdijk, Bertrand Toën
      Pages 127-141
    4. Algebraic stacks

      • Ieke Moerdijk, Bertrand Toën
      Pages 143-158
    5. Simplicial commutative algebras

      • Ieke Moerdijk, Bertrand Toën
      Pages 159-165
    6. Derived stacks and derived algebraic stacks

      • Ieke Moerdijk, Bertrand Toën
      Pages 167-177
    7. Examples of derived algebraic stacks

      • Ieke Moerdijk, Bertrand Toën
      Pages 179-184
  4. Back Matter

    Pages 185-186

About this book

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

  • Mathematisch Instituut, Rijksuniversiteit Utrecht, Utrecht, Netherlands

    Ieke Moerdijk

  • UMR CNRS 5580, Labo. Mathématiques Emile Picard, Université Toulouse III, Toulouse CX 4, France

    Bertrand Toën

Bibliographic Information

Buying options

eBook
USD 24.99
Price excludes VAT (USA)
  • ISBN: 978-3-0348-0052-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 34.95
Price excludes VAT (USA)