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Higher Segal Spaces

  • Tobias Dyckerhoff
  • Mikhail Kapranov
Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 2244)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 1-8
  3. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 9-30
  4. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 31-70
  5. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 71-84
  6. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 85-94
  7. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 95-106
  8. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 107-124
  9. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 125-151
  10. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 153-167
  11. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 169-199
  12. Tobias Dyckerhoff, Mikhail Kapranov
    Pages 201-208
  13. Back Matter
    Pages 209-218

About this book

Introduction

This monograph initiates a theory of new categorical structures that generalize the simplicial Segal property to higher dimensions. The authors introduce the notion of a d-Segal space, which is a simplicial space satisfying locality conditions related to triangulations of d-dimensional cyclic polytopes. Focus here is on the 2-dimensional case. Many important constructions are shown to exhibit the 2-Segal property, including Waldhausen’s S-construction, Hecke-Waldhausen constructions, and configuration spaces of flags. The relevance of 2-Segal spaces in the study of Hall and Hecke algebras is discussed.

Higher Segal Spaces marks the beginning of a program to systematically study d-Segal spaces in all dimensions d. The elementary formulation of 2-Segal spaces in the opening chapters is accessible to readers with a basic background in homotopy theory. A chapter on Bousfield localizations provides a transition to the general theory, formulated in terms of combinatorial model categories, that features in the main part of the book. Numerous examples throughout assist readers entering this exciting field to move toward active research; established researchers in the area will appreciate this work as a reference.

Keywords

Higher category theory Segal Spaces 2-Segal property 2-Segal spaces 1-Segal spaces pre-category object simplicial topological space Waldhausen's S-construction Bousfield localization path space criterion exact ∞-category monads in bispans

Authors and affiliations

  • Tobias Dyckerhoff
    • 1
  • Mikhail Kapranov
    • 2
  1. 1.Department of MathematicsUniversity of HamburgHamburgGermany
  2. 2.Kavli Institute for the Physics and Mathematics of the UniverseKashiwaJapan

Bibliographic information