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From Categories to Homotopy Theory


Part of Cambridge Studies in Advanced Mathematics

  • Publication planned for: April 2020
  • availability: Not yet published - available from April 2020
  • format: Hardback
  • isbn: 9781108479622

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About the Authors
  • Category theory provides structure for the mathematical world and is seen everywhere in modern mathematics. With this book, the author bridges the gap between pure category theory and its numerous applications in homotopy theory, providing the necessary background information to make the subject accessible to graduate students or researchers with a background in algebraic topology and algebra. The reader is first introduced to category theory, starting with basic definitions and concepts before progressing to more advanced themes. Concrete examples and exercises illustrate the topics, ranging from colimits to constructions such as the Day convolution product. Part II covers important applications of category theory, giving a thorough introduction to simplicial objects including an account of quasi-categories and Segal sets. Diagram categories play a central role throughout the book, giving rise to models of iterated loop spaces, and feature prominently in functor homology and homology of small categories.

    • Includes diagrammatical proofs, examples and exercises to encourage an active way of learning
    • Provides enough background from category theory to make advanced topics such as K-theory, iterated loop spaces and functor homology accessible to readers
    • Makes abstract concepts tangible, encouraging readers to learn what can be an intimidating subject
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    Product details

    • Publication planned for: April 2020
    • format: Hardback
    • isbn: 9781108479622
    • length: 400 pages
    • dimensions: 235 x 156 x 26 mm
    • weight: 0.68kg
    • contains: 115 exercises
    • availability: Not yet published - available from April 2020
  • Table of Contents

    Part I. Category Theory:
    1. Basic notions in category theory
    2. Natural transformations and the Yoneda lemma
    3. Colimits and limits
    4. Kan extensions
    5. Comma categories and the Grothendieck construction
    6. Monads and comonads
    7. Abelian categories
    8. Symmetric monoidal categories
    9. Enriched categories
    Part II. From Categories to Homotopy Theory:
    10. Simplicial objects
    11. The nerve and the classifying space of a small category
    12. A brief introduction to operads
    13. Classifying spaces of symmetric monoidal categories
    14. Approaches to iterated loop spaces via diagram categories
    15. Functor homology
    16. Homology and cohomology of small categories

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    From Categories to Homotopy Theory

    Birgit Richter

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  • Author

    Birgit Richter, Universität Hamburg
    Birgit Richter is Professor of Mathematics at Universität Hamburg. She is the co-editor of Structured Ring Spectra (2004) and New Topological Contexts for Galois Theory and Algebraic Geometry (2009).

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