COHERENCE FOR TRICATEGORIES

This work defines the concept of tricategory as the natural 3-dimensio nal generalization of bicategory. Trihomomorphism and triequivalence f or tricategories are also defined so as to extend the concepts of homo morphism and biequivalence for bicategories. The main theorem is a coh erence theorem for tricategories which asserts the existence of a trie quivalence between each tricategory and some V-category, where V is th e category of 2-categories equipped with the strong tenser product of J. W. Gray. Further, it is shown that while not every tricategory is t riequivalent to a 3-category, every tricategory that is locally a 2-ca tegory and whose composition is a 2-functor is triequivalent to a 3-ca tegory. The work has applications to cohomology theory, homotopy theor y, bicategory enriched categories, and bicategories with extra structu re.