The n-Category Café

It has a very definite esthetic which emphasizes thoroughness. Some fundamental results on 2-categories and bicategories have never been given proofs with all the calculations explicitly spelled out. Johnson and Yau aim to correct this. Words like “obviously” are avoided. Read Niles Johnson for more on why they took the approach they did. The idea is that you can skip details if you don’t want to see them — but if you want to see them, you can.

Here are some random comments, nothing like an organized book review:

1. They explain the axiom of universes on page 1. The definition of “universe” is Definition 1.1.1. None of this “ignoring size issues” stuff.

2. They give a careful proof that each pasting diagram in a bicategory defines a unique 2-cell.

3. They spend a chapter proving the “Whitehead Theorem for Bicategories”. Namely: a pseudofunctor of bicategories $F \colon B \to C$ is a biequivalence if and only if $F$ is (1) essentially surjective on objects, (2) essentially full on 1-cells, and (3) fully faithful on 2-cells. The name here comes from the analogy with Whitehead’s Theorem in homotopy theory. They prove the result using something they call “Quillen’s Theorem A for Bicategories”. Again, this not something Quillen proved, but an analogue of something he proved. Quillen’s Theorems A and B gave conditions for a functor between categories to induce a homotopy equivalence (resp. fibration) on the geometric realizations of their nerves.

4. They prove the bicategorical Yoneda Lemma and use it to prove that every bicategory is biequivalent to a 2-category. They do this using their Whitehead Theorem for Bicategories.

5. They study the Grothendieck construction in detail and construct a 2-monad on $\mathsf{Cat}/C$ whose pseudoalgebras are cloven fibrations over $C$. I’d never thought about that. The strict algebras are the split fibrations over $C$. They also discuss a bicategorical version of the Grothendieck construction.

6. They introduce tricategories and the tricategory of bicategories. They do this near the end. They do not use tricategories throughout the book as a tool to study bicategories, just as most introductions to categories do not make heavy use of 2-categories. While experts might enjoy “go higher to soar above the difficulties”, there are obvious problems with this sort of strategy, since to be self-contained you need to explain the $(n+1)$-categorical material before you apply it to the $n$-categorical material… and before you know it you’ll be doing $(\infty,1)$-categories — which may be a good thing, but definitely a different thing than explaining $n$-categories for some fixed low $n$.

7. Near they end they define monoidal, braided, sylleptic and symmetric bicategories, but they draw the coherence laws in a way that make them look like random junk. This is a pity because it hides the fact that the complicated coherence laws governing these structures follow patterns that are combinatorially interesting and visually beautiful when drawn right. I bet that Mike Stay would be glad to share his beautiful diagrams with Johnson and Yau, to make this portion of the book more appealing.

In summary: this book finally provides 2-dimensional category theory with the thorough textbook treatment it deserves.