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Coherence in Three-Dimensional Category Theory

Book Description

Dimension three is an important test-bed for hypotheses in higher category theory and occupies something of a unique position in the categorical landscape. At the heart of matters is the coherence theorem, of which this book provides a definitive treatment, as well as covering related results. Along the way the author treats such material as the Gray tensor product and gives a construction of the fundamental 3-groupoid of a space. The book serves as a comprehensive introduction, covering essential material for any student of coherence and assuming only a basic understanding of higher category theory. It is also a reference point for many key concepts in the field and therefore a vital resource for researchers wishing to apply higher categories or coherence results in fields such as algebraic topology or theoretical computer science.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Introduction
    1. 1. Tricategories
    2. 2. Gray-monads
    3. 3. An outline
      1. Acknowledgements
  7. Part I: Background
    1. 1. Bicategorical background
      1. 1.1 Bicategorical conventions
      2. 1.2 Mates in bicategories
    2. 2. Coherence for bicategories
      1. 2.1 The Yoneda embedding
      2. 2.2 Coherence for bicategories
      3. 2.3 Coherence for functors
    3. 3. Gray-categories
      1. 3.1 The Gray tensor product
      2. 3.2 Cubical functors
      3. 3.3 The monoidal category Gray
      4. 3.4 A factorization
  8. Part II: Tricategories
    1. 4. The algebraic definition of tricategory
      1. 4.1 Basic definition
      2. 4.2 Adjoint equivalences and tricategory axioms
      3. 4.3 Trihomomorphisms and other higher cells
      4. 4.4 Unpacked versions
      5. 4.5 Calculations in tricategories
      6. 4.6 Comparing definitions
    2. 5. Examples
      1. 5.1 Primary example: Bicat
      2. 5.2 Fundamental 3-groupoids
    3. 6. Free constructions
      1. 6.1 Graphs
      2. 6.2 The category of tricategories
      3. 6.3 Free Gray-categories
    4. 7. Basic structure
      1. 7.1 Structure of functors
      2. 7.2 Structure of transformations
      3. 7.3 Pseudo-icons
      4. 7.4 Change of structure
      5. 7.5 Triequivalences
    5. 8. Gray-categories and tricategories
      1. 8.1 Cubical tricategories
      2. 8.2 Gray-categories
    6. 9. Coherence via Yoneda
      1. 9.1 Local structure
      2. 9.2 Global results
      3. 9.3 The cubical Yoneda lemma
      4. 9.4 Coherence for tricategories
    7. 10. Coherence via free constructions
      1. 10.1 Coherence for tricategories
      2. 10.2 Coherence and diagrams of constraints
      3. 10.3 A non-commuting diagram
      4. 10.4 Strictifying tricategories
      5. 10.5 Coherence for functors
      6. 10.6 Strictifying functors
  9. Part III: Gray-monads
    1. 11. Codescent in Gray-categories
      1. 11.1 Lax codescent diagrams
      2. 11.2 Codescent diagrams
      3. 11.3 Codescent objects
    2. 12. Codescent as a weighted colimit
      1. 12.1 Weighted colimits in Gray-categories
      2. 12.2 Examples: coinserters and coequifiers
      3. 12.3 Codescent
    3. 13. Gray-monads and their algebras
      1. 13.1 Enriched monads and algebras
      2. 13.2 Lax algebras and their higher cells
      3. 13.3 Total structures
    4. 14. The reflection of lax algebras into strict algebras
      1. 14.1 The canonical codescent diagram of a lax algebra
      2. 14.2 The left adjoint, lax case
      3. 14.3 The left adjoint, pseudo case
    5. 15. A general coherence result
      1. 15.1 Weak codescent objects
      2. 15.2 Coherence for pseudo-algebras
  10. Bibliography
  11. Index