In this chapter, we will explicitly construct two example tricategories. One of these examples will be useful later, and the other is just basic, but important. The main common feature of these examples is that they can, without much effort, be constructed directly, without any sophisticated understanding of tricategories.

Primary example: Bicat

This section will establish two key results. The basic result is that the collection of bicategories, functors, transformations, and modifications forms a tricategory. This will be shown directly by calculation. Later, we will also be able prove it by transporting the tricategory structure from the tricategory Gray. This will also prove that Bicat is triequivalent to an easily determined full sub-Gray-category of Gray which we call Gray′.

It should also be noted that there are two natural tricategory structures on the collection of bicategories, functor, transformations, and modification; this becomes clear when defining the horizontal composite of transformations, as there are two obvious choices and a canonical comparison map between them. This bifurcation will be noted, but it will not be important to the theory developed here. The line of proof followed here is largely calculational.

The first piece of data we must construct is the hom-bicategory Bicat(A, B) for bicategories A and B. It has objects the functors F : A → B, 1-cells the transformations α: F⇒ G, and 2-cells the modifications Γ: α⇛ β.