Review of Coherence in Three - Dimensional Category Theory

Nick Gurski’s new book addresses some central concerns of the subject known as higher category theory; and yet, it is some distance from what many people now understand by that term. This may puzzle some. I will therefore begin by locating Gurski’s book within the mathematical landscape. Let n ∈ N ∪ {∞}. Roughly, an n-category consists of some objects, some 1-morphisms between objects, some 2-morphisms between 1-morphisms (when those 1-morphisms have the same domain and codomain), and so on, up to n-morphisms between (n − 1)-morphisms, or without end if n = ∞. These morphisms can be composed in various ways, and composition satisfies axioms. Thus, a 0-category is a set and a 1-category is a category. The devil is in the detail. If we ask that the various compositions satisfy strict axioms, such as h ◦ (g ◦ f) = (h ◦ g) ◦ f , then we arrive at the definition of a so-called strict n-category. These are very well understood, but the definition excludes many natural examples. For instance, given a topological space X and n ∈ N ∪ {∞}, we would like there to be an n-category Πn(X) in which the objects are the points of X, the 1-morphisms are paths, 2-morphisms are homotopies between paths, 3-morphisms are homotopies between homotopies, and so on. But concatenation of homotopies is not strictly associative or unital, so Πn(X) is not a strict n-category. We are therefore led to seek a definition of non-strict, or weak, n-category, that includes such examples. This is where the landscape opens up. On one side, there are algebraic approaches to the problem. Here, an n-category is conceived as an algebraic structure, consisting of a collection of morphisms of each dimension, equipped with various operations satisfying universally quantified equations. On the other side, the non-algebraic approach does not attempt to assign a definite composite to each composable pair of morphisms, but merely asserts the existence of some third morphism satisfying a suitable universal property. This distinction can be explained by analogy with cartesian products of sets. One person might take the approach that any two sets X and Y have a definite product X × Y ; but it should then be observed that the products X × (Y ×Z) and (X × Y ) × Z are not actually equal, only canonically isomorphic, and that, moreover, these isomorphisms satisfy equations of their own (such as a pentagonal identity for four-fold products). A different person might assert that the product is only defined up to isomorphism; but then they need to state its characterizing universal property, and they lose the right to speak of a specific set called X × Y , at least without further justification. The algebraic approach goes back half a century, to Bénabou’s definition of bicategory (weak 2-category) [1]. His work made plain one difficulty of this approach: in the definitions of bicategory, functor between bicategories, and so on, the coherence axioms (such as the aforementioned pentagon) are quite complicated. The complications multiply in dimension 3, as demonstrated by the