### 2-Dimensional Categories

#### Posted by John Baez

There’s a comprehensive introduction to 2-categories and bicategories now, free on the arXiv:

- Niles Johnson and Donald Yau,
*2-Dimensional Categories*, 476 pages.

Abstract.This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

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:

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.

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

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.

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.

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.

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 $n$.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.

## Re: 2-Dimensional Categories

Hi all, and thanks John for taking the time to think and write about the content! We’re even grateful for the fair criticism about those diagrams – your point that the combinatorial patterns could be more evident is well-taken.

As is perhaps inevitable in a project like this, a couple of other people have also pointed out minor errors. While unfortunate, we would

muchprefer to correct such things at this stage, rather than issue errata after the book appears in print. So if you’ve noticed anything else, do let us know either here or elsewhere.