Higher-dimensional category theory involves categorical structures that, in a sense, operate at several levels. The simplest cases are 2-categories, which add a morphism layer to an ordinary category. Think of morphisms between the layer-1 morphisms, that is, the usual morphisms between objects. As usual, certain axioms have to be satisfied. Higher-dimensional categories are defined by extending this basic idea further.

This book investigates the third dimension in this dimensional tower of categories. The text begins with a short introduction that outlines the book and presents its main concepts: tricategories and Gray monads. The main body of the book is divided into three parts.

Part 1 recaps some basic definitions and concepts of the theory of bicategories; proves a bicategorical version of the Yoneda lemma, together with some coherence results following from the lemma; and introduces Gray categories, which are 2-categories endowed with a tensor product between the 2-categories.

Tricategories are the main topic of the second part, which introduces the subject and proves several of their properties. This part culminates with a tricategorical Yoneda lemma, together with coherence results following from the lemma. An important aspect of this part is the deep relationship between tricategories and Gray categories.

The last part turns to Gray monads and certain limit constructions in Gray categories, concluding with coherence results with respect to pseudo-algebras.

The book is mathematically sound, although very abstract and therefore very hard to read for someone not deeply involved with its subject. It is not well suited for getting into the subject since it is often assumed that the reader is already familiar with certain standard texts.

However, for a reader knowledgeable in the field, the author has presented interesting results and technical approaches that use connections between some category theory concepts that are not commonly known. Although I was often left out (because I have not read the standard texts), I got a glimpse of what is going on in this area of mathematics. Some guidance for an intuitive understanding would have been helpful, but this is probably difficult for such highly abstract topics.