Bicat Is Not Triequivalent to Gray

Bicat is the tricategory of bicategories, homomorphisms, pseudonatural transformations, and modifications. Gray is the subtricategory of 2-categories, 2functors, pseudonatural transformations, and modifications. We show that these two tricategories are not triequivalent. 1. Background. Weakening the notion of 2-category by replacing all equations between 1-cells by suitably coherent isomorphisms gives the notion of bicategory [1]. The analogous weakening of a 2-functor is called a homomorphism of bicategories, and the weakening of a 2-natural transformation is a pseudonatural transformation. There are also modifications between 2-natural or pseudonatural transformations, but this notion does not need to be weakened. The bicategories, homomorphisms, pseudonatural transformations, and modifications form a tricategory (a weak 3-category) called Bicat. The subtricategory of Bicat containing only the 2-categories as objects, and only the 2-functors as 1-cells, but with all 2-cells and 3-cells between them, is called Gray. As well as being a particular tricategory, there is another important point of view on Gray. The category 2-Cat of 2-categories and 2-functors is cartesian closed, but it also has a different symmetric monoidal closed structure [3], for which the internal hom [A ,B] is the 2-category of 2-functors, pseudonatural transformations, and modifications between A and B. A category enriched over 2-Cat with respect to this closed structure is called a Gray-category. A Gray-category has 2-categories as hom-objects, so is a 3-dimensional categorical structure, and it can be seen as a particular sort of tricategory. The closed structure of 2-Cat gives it a canonical enrichment over itself and the resulting Graycategory is just Gray. Gray is also sometimes used as a name for 2-Cat with this monoidal structure. A homomorphism of bicategories T : A → C is called a biequivalence if it induces equivalences TA,B : A (A,B) → B(TA, TB) of hom-categories for all objects A,B ∈ C (T is locally an equivalence), and every object C ∈ C is equivalent in C to one of the form TA (T is biessentially surjective on objects). We then write A ∼ B. Every bicategory is equivalent to a 2-category [5]. A trihomomorphism of tricategories T : A → C is called a triequivalence if it induces biequivalences TA,B : A (A,B) → B(TA, TB) of hom-bicategories for all objects A,B ∈ A (T is locally a biequivalence), and every object C ∈ C is biequivalent in C to one The support of the Australian Research Council and DETYA is gratefully acknowledged. Received by the editors 2006-12-11 and, in revised form, 2007-01-05. Transmitted by Ross Street. Published on 2007-01-08. 2000 Mathematics Subject Classification: 18D05.