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Coherence for Tricategories
R. Gordon, A. J. Power, and Ross Street

Memoirs of the American Mathematical Society
1995; 81 pp; softcover
Volume: 117
ISBN-10: 0-8218-0344-1
ISBN-13: 978-0-8218-0344-8
List Price: US$39
Individual Members: US$23.40
Institutional Members: US$31.20
Order Code: MEMO/117/558
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The need to address the appropriate three-dimensional generalization of category (tricategory) has been felt in homotopy theory, low-dimensional topology, cohomology theory, category theory, and quantum field theory. Benabou's bicategories provide the two-dimensional notion into which examples naturally fit. In developing the theory of bicategories it is very reassuring to know the coherence theorem: They can be replaced by the stricter 2-categories which are merely categories enriched in the category of categories.

In this book, the authors provide ...

  • the unique source of the full definition of tricategory
  • a thorough and complete proof of the coherence theorem for tricategories
  • a wholly modern source of material on Gray's tensor product of 2-categories


Research mathematicians, theoretical physicists, algebraic topologists, 3-D computer scientists, and theoretical computer scientists.

Table of Contents

  • Introduction
  • The definition of tricategory
  • Trihomomorphisms, triequivalence, and \(\operatorname{\mathbf{Tricat}}(T,S)\)
  • Cubical functors and tricategories, and the monoidal category \(\operatorname{\mathbf{Gray}}\)
  • \(\operatorname{\mathbf{Gray}}\)-categories, and \(\operatorname{\mathbf{Bicat}}\) as a tricategory
  • The \(\operatorname{\mathbf{Gray}}\)-category \(\operatorname{Prep(T)}\) of prerepresentations of \(T\)
  • The "Yoneda Embedding"
  • The Main Theorem
  • Acknowledgements
  • References
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