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Categorical Homotopy Theory

Book Description

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.

Table of Contents

  1. Cover
  2. Half title
  3. Series
  4. Title
  5. Copyright
  6. Dedication
  7. Epigraph
  8. Table of Contents
  9. Preface
  10. I Derived functors and homotopy (co)limits
    1. 1 All concepts are Kan extensions
      1. 1.1 Kan extensions
      2. 1.2 A formula
      3. 1.3 Pointwise Kan extensions
      4. 1.4 All concepts
      5. 1.5 Adjunctions involving simplicial sets
    2. 2 Derived functors via deformations
      1. 2.1 Homotopical categories and derived functors
      2. 2.2 Derived functors via deformations
      3. 2.3 Classical derived functors between abelian categories
      4. 2.4 Preview of homotopy limits and colimits
    3. 3 Basic concepts of enriched category theory
      1. 3.1 A first example
      2. 3.2 The base for enrichment
      3. 3.3 Enriched categories
      4. 3.4 Underlying categories of enriched categories
      5. 3.5 Enriched functors and enriched natural transformations  
      6. 3.6 Simplicial categories
      7. 3.7 Tensors and cotensors
      8. 3.8 Simplicial homotopy and simplicial model categories
    4. 4 The unreasonably effective (co)bar construction
      1. 4.1 Functor tensor products
      2. 4.2 The bar construction
      3. 4.3 The cobar construction
      4. 4.4 Simplicial replacements and colimits
      5. 4.5 Augmented simplicial objects and extra degeneracies
    5. 5 Homotopy limits and colimits: The theory
      1. 5.1 The homotopy limit and colimit functors
      2. 5.2 Homotopical aspects of the bar construction
    6. 6 Homotopy limits and colimits: The practice
      1. 6.1 Convenient categories of spaces
      2. 6.2 Simplicial model categories of spaces
      3. 6.3 Warnings and simplifications
      4. 6.4 Sample homotopy colimits
      5. 6.5 Sample homotopy limits
      6. 6.6 Homotopy colimits as weighted colimits
  11. II Enriched homotopy theory
    1. 7 Weighted limits and colimits
      1. 7.1 Weighted limits in unenriched category theory
      2. 7.2 Weighted colimits in unenriched category theory
      3. 7.3 Enriched natural transformations and enriched ends
      4. 7.4 Weighted limits and colimits
      5. 7.5 Conical limits and colimits
      6. 7.6 Enriched completeness and cocompleteness
      7. 7.7 Homotopy (co)limits as weighted (co)limits
      8. 7.8 Balancing bar and cobar constructions
    2. 8 Categorical tools for homotopy (co)limit computations
      1. 8.1 Preservation of weighted limits and colimits
      2. 8.2 Change of base for homotopy limits and colimits
      3. 8.3 Final functors in unenriched category theory
      4. 8.4 Final functors in enriched category theory
      5. 8.5 Homotopy final functors
    3. 9 Weighted homotopy limits and colimits
      1. 9.1 The enriched bar and cobar construction
      2. 9.2 Weighted homotopy limits and colimits
    4. 10 Derived enrichment
      1. 10.1 Enrichments encoded as module structures
      2. 10.2 Derived structures for enrichment
      3. 10.3 Weighted homotopy limits and colimits, revisited
      4. 10.4 Homotopical structure via enrichment
      5. 10.5 Homotopy equivalences versus weak equivalences
  12. III Model categories and weak factorization systems
    1. 11 Weak factorization systems in model categories
      1. 11.1 Lifting problems and lifting properties
      2. 11.2 Weak factorization systems
      3. 11.3 Model categories and Quillen functors
      4. 11.4 Simplicial model categories
      5. 11.5 Weighted colimits as left Quillen bifunctors
    2. 12 Algebraic perspectives on the small object argument
      1. 12.1 Functorial factorizations
      2. 12.2 Quillen’s small object argument
      3. 12.3 Benefits of cofibrant generation
      4. 12.4 Algebraic perspectives
      5. 12.5 Garner’s small object argument
      6. 12.6 Algebraic weak factorization systems and universal properties
      7. 12.7 Composing algebras and coalgebras
      8. 12.8 Algebraic cell complexes
      9. 12.9 Epilogue on algebraic model categories
    3. 13 Enriched factorizations and enriched lifting properties
      1. 13.1 Enriched arrow categories
      2. 13.2 Enriched functorial factorizations
      3. 13.3 Enriched lifting properties
      4. 13.4 Enriched weak factorization systems
      5. 13.5 Enriched model categories
      6. 13.6 Enrichment as coherence
    4. 14 A brief tour of Reedy category theory
      1. 14.1 Latching and matching objects
      2. 14.2 Reedy categories and the Reedy model structures
      3. 14.3 Reedy cofibrant objects and homotopy (co)limits
      4. 14.4 Localizations and completions of spaces
      5. 14.5 Homotopy colimits of topological spaces
  13. IV Quasi-categories
    1. 15 Preliminaries on quasi-categories
      1. 15.1 Introducing quasi-categories
      2. 15.2 Closure properties
      3. 15.3 Toward the model structure
      4. 15.4 Mapping spaces
    2. 16 Simplicial categories and homotopy coherence
      1. 16.1 Topological and simplicial categories
      2. 16.2 Cofibrant simplicial categories are simplicial computads
      3. 16.3 Homotopy coherence
      4. 16.4 Understanding the mapping spaces CX(x, y)
      5. 16.5 A gesture toward the comparison
    3. 17 Isomorphisms in quasi-categories
      1. 17.1 Join and slice
      2. 17.2 Isomorphisms and Kan complexes
      3. 17.3 Inverting simplices
      4. 17.4 Marked simplicial sets
      5. 17.5 Inverting diagrams of isomorphisms
      6. 17.6 A context for invertibility
      7. 17.7 Homotopy limits of quasi-categories
    4. 18 A sampling of 2-categorical aspects of quasi-category theory
      1. 18.1 The 2-category of quasi-categories
      2. 18.2 Weak limits in the 2-category of quasi-categories
      3. 18.3 Arrow quasi-categories in practice
      4. 18.4 Homotopy pullbacks
      5. 18.5 Comma quasi-categories
      6. 18.6 Adjunctions between quasi-categories
      7. 18.7 Essential geometry of terminal objects
  14. Bibliography
  15. Glossary of Notation
  16. Index