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Categorical Homotopy Theory

Part of New Mathematical Monographs

  • Date Published: August 2014
  • availability: Available
  • format: Hardback
  • isbn: 9781107048454


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About the Authors
  • This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Emily Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence.

    • Gives a unified presentation of the theory of homotopy limits and colimits
    • Isolates the key categorical components of the definition of a model category
    • Discusses the enriched category theory relevant to homotopy theory
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    Product details

    • Date Published: August 2014
    • format: Hardback
    • isbn: 9781107048454
    • length: 372 pages
    • dimensions: 229 x 152 x 25 mm
    • weight: 0.72kg
    • contains: 55 exercises
    • availability: Available
  • Table of Contents

    Part I. Derived Functors and Homotopy (Co)limits:
    1. All concepts are Kan extensions
    2. Derived functors via deformations
    3. Basic concepts of enriched category theory
    4. The unreasonably effective (co)bar construction
    5. Homotopy limits and colimits: the theory
    6. Homotopy limits and colimits: the practice
    Part II. Enriched Homotopy Theory:
    7. Weighted limits and colimits
    8. Categorical tools for homotopy (co)limit computations
    9. Weighted homotopy limits and colimits
    10. Derived enrichment
    Part III. Model Categories and Weak Factorization Systems:
    11. Weak factorization systems in model categories
    12. Algebraic perspectives on the small object argument
    13. Enriched factorizations and enriched lifting properties
    14. A brief tour of Reedy category theory
    Part IV. Quasi-Categories:
    15. Preliminaries on quasi-categories
    16. Simplicial categories and homotopy coherence
    17. Isomorphisms in quasi-categories
    18. A sampling of 2-categorical aspects of quasi-category theory.

  • Author

    Emily Riehl, Harvard University, Massachusetts
    Emily Riehl is a Benjamin Peirce Fellow in the Department of Mathematics at Harvard University, Massachusetts and a National Science Foundation Mathematical Sciences Postdoctoral Research Fellow.

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