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February 17, 2020

2-Dimensional Categories

Posted by John Baez

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

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:

  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:BCF \colon B \to C is a biequivalence if and only if FF 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 Cat/C\mathsf{Cat}/C whose pseudoalgebras are cloven fibrations over CC. I’d never thought about that. The strict algebras are the split fibrations over CC. 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)(n+1)-categorical material before you apply it to the nn-categorical material… and before you know it you’ll be doing (,1)(\infty,1)-categories — which may be a good thing, but definitely a different thing than explaining nn-categories for some fixed low nn.

  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.

Posted at February 17, 2020 3:44 PM UTC

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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 much prefer 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.

Posted by: Niles Johnson on February 18, 2020 4:35 PM | Permalink | Reply to this

Re: 2-Dimensional Categories

Since you said that you prefer to receive comments before the book goes to print: is there already a rough date for that? I ask because I plan to read it. I have only read some parts so far, but apparently this book is something I always wanted but didn’t know that I want (and also need) it for my research.

Posted by: Martin Brandenburg on February 18, 2020 10:10 PM | Permalink | Reply to this

Re: 2-Dimensional Categories

Oh, no idea of a date at this stage. We only submitted it to a publisher after putting it on the arxiv, so I expect it will be a while.

Very glad you’re finding it useful! :)

Posted by: Niles Johnson on February 19, 2020 1:17 AM | Permalink | Reply to this

Re: 2-Dimensional Categories

I like the fact that set-theoretic universes are used, and mentioned early. (I recently gave a talk at a Set Theory UK meeting where I advocated using universes more widely.)

However, what is written on page 1 is on the face of it contradictory. Every set belongs to a universe, which is itself a set, and then a set is defined to be an element of a specific universe U. I know this is a redefinition, and is exactly the conventional approach. However, I am not sure it would be transparent to someone who does not already know what is going on.

In some cases you must want to deal with large categories, for example the bicategory of categories, and consider the functor category of all functors from Group to Set, for example. This is a set of classes, so you want to be able to treat classes like sets. The Universes axiom does allow you to do that (at least in conjunction with enough other axioms!) but the convention given does not explain this issue.

So to me it appears that mentioning universes is really hiding the fact that something is being brushed under the carpet. I would suggest saying a little bit more here, for example: 1) giving all the axioms you need for sets, 2) explaining that classes are sets in a larger universe and so can be manipulated like sets 3) giving at least one reference at this point rather than leaving them to p23.

Posted by: Jonathan Kirby on February 21, 2020 8:11 PM | Permalink | Reply to this

Re: 2-Dimensional Categories

The axiom of universes says there is an unbounded class of universes, since every universe is then contained in a large one.

More egregious is the error in the definition of universe (I already pinged Niles and Donald on this), which apparently was copied from Borceux’s Handbook of Categorical Algebra I (Definition 1.1.2).

Posted by: David Roberts on February 22, 2020 5:53 AM | Permalink | Reply to this

Re: 2-Dimensional Categories

For what it’s worth, a corrected version of this definition is given in Streicher’s notes Introduction to CATEGORY THEORY and CATEGORICAL LOGIC, in Definition 6.1.

Posted by: David Roberts on February 23, 2020 11:15 PM | Permalink | Reply to this

Re: 2-Dimensional Categories

Indeed, thanks both Jonathan and David for the feedback. Along with fixing the error David mentioned, we also added a short result about some basic constructions.

Since the book is not intended to be an introduction to 1-category theory, and Chapter 1 is only a concise review, I hope you will forgive us for not adding too much more about the set-theoretic foundations.

Posted by: Niles Johnson on February 24, 2020 8:13 PM | Permalink | Reply to this

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