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A Handbook of Model Categories

  • Book
  • © 2021

Overview

Authors:
  1. Scott Balchin

    1. Max Planck Institute for Mathematics, Bonn, North-Rhine-Westphalia, Germany

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  • Highlights the key points of the theory before discussing various properties and structures that a model category may have
  • Boasts over 200 pages of examples of model categories in algebra and topology, and everything in between, ranging over 12 themes
  • Features a table of curiosities highlighting a range of, sometimes unexpected, phenomena that occur when working with model categories

Part of the book series: Algebra and Applications (AA, volume 27)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xv
    Download chapter PDF

  2. Introduction

    • Scott Balchin
    Pages 1-5

  3. The Theory and Practice of Model Categories

    1. Front Matter

      Pages 7-9
      Download chapter PDF

    2. On Quillen Model Categories

      • Scott Balchin
      Pages 11-28

    3. Properties

      • Scott Balchin
      Pages 29-40

    4. New Models from Old Ones

      • Scott Balchin
      Pages 41-88

    5. Relation to \((\infty ,1)\)-Categories

      • Scott Balchin
      Pages 89-95

  4. Examples

    1. Front Matter

      Pages 97-100
      Download chapter PDF

    2. Simplicial Sets

      • Scott Balchin
      Pages 101-130

    3. Topological Spaces

      • Scott Balchin
      Pages 131-143

    4. Chain Complexes

      • Scott Balchin
      Pages 145-159

    5. Categories

      • Scott Balchin
      Pages 161-177

    6. Spectra

      • Scott Balchin
      Pages 179-193

    7. Simplicial Categories

      • Scott Balchin
      Pages 195-203

    8. Bisimplicial Sets

      • Scott Balchin
      Pages 205-215

    9. Relative Categories

      • Scott Balchin
      Pages 217-225

    10. Dendroidal Sets

      • Scott Balchin
      Pages 227-245

    11. Cyclic Sets

      • Scott Balchin
      Pages 247-259

    12. \(C^*\)-Algebras

      • Scott Balchin
      Pages 261-276

    13. Miscellanea

      • Scott Balchin
      Pages 277-300
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Keywords

About this book

This book outlines a vast array of techniques and methods regarding model categories, without focussing on the intricacies of the proofs. Quillen model categories are a fundamental tool for the understanding of homotopy theory. While many introductions to model categories fall back on the same handful of canonical examples, the present book highlights a large, self-contained collection of other examples which appear throughout the literature. In particular, it collects a highly scattered literature into a single volume.
The book is aimed at anyone who uses, or is interested in using, model categories to study homotopy theory. It is written in such a way that it can be used as a reference guide for those who are already experts in the field. However, it can also be used as an introduction to the theory for novices.



Reviews

“This book is a catalog of definitions, theorems, examples and counterexamples in model categories. … It is written both for beginners in this area of mathematics and for advanced researchers. In this book, the beginner will find an overview of the results in this area without being overwhelmed by technical details. The advanced researcher will be interested in having so many theorems and examples and counterexamples gathered in the same place.” (Philippe Gaucher, Mathematical Reviews, November, 2022)

“This book is well-written and divided into eighteen chapters … . it can be said that this book will be one of the cult books about model categories in the literature, with its rich, diverse and current examples.” (Haydar Can Kaya, zbMATH 1486.18001, 2022)

Authors and Affiliations

  • Max Planck Institute for Mathematics, Bonn, North-Rhine-Westphalia, Germany

    Scott Balchin

About the author

Scott Balchin currently holds a position at the Max Planck Institute for Mathematics, Germany. Previously he was a postdoctoral research fellow at the University of Warwick, England. He has published several articles on the use of Quillen model categories in various aspects of homotopy theory.

Bibliographic Information

  • Book Title: A Handbook of Model Categories

  • Authors: Scott Balchin

  • Series Title: Algebra and Applications

  • DOI: https://doi.org/10.1007/978-3-030-75035-0

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2021

  • Hardcover ISBN: 978-3-030-75034-3Published: 30 October 2021

  • Softcover ISBN: 978-3-030-75037-4Published: 30 October 2022

  • eBook ISBN: 978-3-030-75035-0Published: 29 October 2021

  • Series ISSN: 1572-5553

  • Series E-ISSN: 2192-2950

  • Edition Number: 1

  • Number of Pages: XV, 326

  • Number of Illustrations: 13 b/w illustrations

  • Topics: Category Theory, Homological Algebra

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