7 - Basic structure

Summary

This chapter will be devoted to studying some aspects of the total algebraic structure consisting of tricategories, functors, transformations, modifications, and perturbations. This chapter will only establish some basic properties that will be used later. There should be a weak 4-category Tricat, but constructing the entire structure would involve a substantial investment, much of which we will not need for the purposes of proving coherence. Since we will be proving a version of the Yoneda lemma for cubical tricategories, we will need to construct functor tricategories of the form [T^op, Gray]. This functor tricategory would be the hom-tricategory in the putative construction of Tricat, but we only construct this in the special case when the target is a Gray-category, and this restriction greatly simplifies many of the calculations. The Yoneda embedding for cubical tricategories will be constructed in Chapter 9; for now we will focus on some basic composition formulas that will be required later.

The third section of this chapter proves some basic results using the notion of pseudo-icons as originally defined by Garner and Gurski (2009). It is often useful to employ the simpler pseudo-icons than transformations in the proofs leading up to coherence. The reason is that many of these transformations have identity components, and much of the data defining those transformations is made more complicated by the insertion of unit constraints because of those identity components.