In the study of higher categories, dimension three occupies an interesting position on the landscape of higher dimensional category theory. From the perspective of a “hands-on” approach to defining weak n-categories, tricategories represent the most complicated kind of higher category that the community at large seems comfortable working with. On the other hand, dimension three is the lowest dimension in which strict n-categories are genuinely more restrictive than fully weak ones, so tricategories should be a sort of jumping off point for understanding general higher dimensional phenomena. This work is intended to provide an accessible introduction to coherence problems in three-dimensional category.

Tricategories

Tricategories were first studied by Gordon, Power, and Street in their 1995 AMSMemoir. They were aware that strict 3-groupoids do not model homotopy 3-types, and thus the aim of their work was to create an explicit definition of a weak 3-category which would not be equivalent (in the appropriate three-dimensional sense) to that of a strict 3-category. The main theorem of Gordon et al. (1995) is often stated: every tricategory is triequivalent to a Gray-category. Triequivalence is a straightforward generalization of the usual notion that two categories are equivalent when there is a functor between them which is essentially surjective, full, and faithful. The new and interesting feature of this result is the appearance of Gray-categories. These are categories which are enriched over the monoidal category Gray; this monoidal category has the category of 2-categories and strict 2-functors as its underlying category, but its monoidal structure is not the Cartesian one.