## Abstract

This work defines the concept of tricategory as the natural 3-dimensional generalization of bicategory. Trihomomorphism and triequivalence for tricategories are also defined so as to extend the concepts of homomorphism and biequivalence for bicategories. The main theorem is a coherence theorem for tricategories which asserts the existence of a triequivalence between each tricategory and some V-category, where V is the category of 2-categories equipped with the strong tenser product of J. W. Gray. Further, it is shown that while not every tricategory is triequivalent to a 3-category, every tricategory that is locally a 2-category and whose composition is a 2-functor is triequivalent to a 3-category. The work has applications to cohomology theory, homotopy theory, bicategory enriched categories, and bicategories with extra structure.

Original language | English |
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Place of Publication | Providence, RI |

Publisher | American Mathematical Society |

Number of pages | 81 |

Volume | 117 |

Edition | 558 |

ISBN (Print) | 9780821803448, 1470401371 |

DOIs | |

Publication status | Published - Sep 1995 |

## Keywords

- TRICATEGORY
- MONOIDAL BICATEGORY
- GRAY-CATEGORY
- ALGEBRAIC HOMOTOPY 3-TYPE
- LAX FUNCTOR OF TRICATEGORIES
- TRIHOMOMORPHISM
- TRIEQUIVALENCE
- CUBICAL FUNCTOR
- CUBICAL TRICATEGORY
- PREREPRESENTATION
- BICATEGORIES
- CATEGORIES