This chapter will develop the basic tools necessary to construct free tricategories and free Gray-categories. First we must decide on the underlying data from which a tricategory is to be generated freely. Second, we must construct both the free tricategory and the free Gray-category on this data. This requires a bit of care as one must pay careful attention to how the universal property is stated; the issue here is that, as we will see, the category of tricategories and strict maps has to be constructed directly, and not as a sub-object of some structure involving more general 1-cells. Finally, we prove some results analogous to those leading up to the proof of the coherence theorem for bicategories.

Graphs

The first step in producing a free tricategory is to decide from what data we will generate such a tricategory. The natural choice is that of a bicategory-enriched graph, but we wish to construct free Gray-categories as well and so we must also work with category-enriched 2-graphs.

Definition 6.1 1. A category-enriched 2-graph X consists of a directed graph X1 ⇉ X0 along with, for each pair of parallel arrows f, g in X1, a category X(f, g). The category of category-enriched 2-graphs, written 2Gr(Cat), has for morphisms X → X′ the pairs (P, F), where P is a map of the underlying directed graphs and F is a collection of functors Ff,g: X(f, g) → X′(P f, Pg).