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.
• A key reference containing background material for a wide variety of relevant topics • Self-contained treatment of all the main results • Includes important explicit examples of tricategories, which will be useful for applications in other fields
Contents
Introduction; Part I. Background: 1. Bicategorical background; 2. Coherence for bicategories; 3. Gray-categories; Part II. Tricategories: 4. The algebraic definition of tricategory; 5. Examples; 6. Free constructions; 7. Basic structure; 8. Gray-categories and tricategories; 9. Coherence via Yoneda; 10. Coherence via free constructions; Part III. Gray monads: 11. Codescent in Gray-categories; 12. Codescent as a weighted colimit; 13. Gray-monads and their algebras; 14. The reflection of lax algebras into strict algebras; 15. A general coherence result; Bibliography; Index.
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