# 2-category theory

I know that we can do a lot of 2-category theory, seeing 2-categories as Cat-enriched categories. Yet, I know that there are some limitations of this approach. I also know that there are many articles which could help to understand 2-category theory... (I am only familiar with a few of the Lack's, Street's and Kelly's articles so far, but I know there are many more important articles). But always when I'm trying to deal seriously with 2-categories, I end finding serious difficulties. So I am sure I have to improve my 2-category theory knowledge in general. And just recently I have become aware of the Gray's Formal Category Theory. So my question is about basics of 2-category theory: which set of articles or books could be considered as a solid base to start thinking seriously about 2-category theory?

I am just trying to avoid a common situation for me: being stuck in a well known (and basic) subject of 2-category theory, ignoring the existence of (classical) literature about this subject.

• What aspect of 2-category theory is troubling you specifically? Is it the theory of 2-categorical or bicategorical limits? Jan 4, 2015 at 14:11
• Now, It is about Kan extensions and quasi-Kan extensions!! However, I had problems on bicategorical limits before - so far, I could solve using some random articles! But what would be the reference you had in mind about bicategorical/2-categorical limits? Jan 4, 2015 at 14:36
• Have you looked at the nLab? This might get you started: ncatlab.org/nlab/search?query=kan+extension Jan 4, 2015 at 14:56
• Thank you. I will take a better look. However I couldn't find anything in nLab before: the problem is the relation of Gray's quasi- Kan extension of a 2-functor and the (strict/enriched) Kan extension of the same 2-functor. Or, more generally, I am trying to understand weak notions of Kan extensions and their relations with the (strict) Kan extension! However, as I said, I noticed my recurrent difficulty in 2-category theory - then I decided to ask here about the literature: for this concern, your answer (as David White's answer) helps a lot my lack of literature knowledge. Thank you Jan 4, 2015 at 15:13

One aspect of 2-category theory which I've sometimes found difficult or tricky is 2-limits (or variants thereof). If that is troubling you too, some of these papers (mentioned in the nLab article on 2-limits) could be helpful:

• Ross Street, Limits indexed by category-valued 2-functors, Journal of Pure and Applied Algebra, Volume 8 No. 2 (June 1976), 149–181. link

• Max Kelly, Elementary observations on 2-categorical limits, Bulletin of the Australian Mathematical Society (1989), 39: 301-317, link.

• Ross Street, Fibrations in Bicategories, Cahiers de topologie et géométrie différentielle catégoriques, tome 21, no. 2 (1980), p. 111-160. numdam pdf. See also the Correction (same journal, Vol. 28 No. 1 (1987), 53-56). link

• Steve Lack, A 2-categories companion arXiv:math.CT/0702535 (see section 6, page 37).

• G.J. Bird, G.M. Kelly, A.J. Power, R.H. Street, Flexible limits for 2-categories, Journal of Pure and Applied Algebra, Vol. 61 No. 1, (November 1989), 1–27. link

• Thomas Fiore, Pseudo Limits, Biadjoints, and Pseudo Algebras, arXiv:math/0408298; see chapters 3, 4, 5.

• John Power, 2-categories, BRICS Notes Series NS-98-7, ISSN 0909-3206 (August 1998). pdf

I would start with the last chapter of Mac Lane's Categories for the Working Mathematician. After that, I'd read Steve Lack's expository article "A 2-Categories Companion." The people you name are the people I'd think of in this field, except I might add Kelly too. Steve Lack has probably written the most on the subject, and I think he writes very well. You can find a complete listing of his writings here. I imagine his students and others coming out of Macquarie will have written a lot about 2-categories in their PhD theses, so you might consider reading some of those to gain expertise with the tools in that field.

It might help to also think a bit about higher categories in general. You could read Towards Higher Categories (edited by John Baez and Peter May), and Carlos Simpson has written a lot on the subject, as well as a very readable textbook called Homotopy Theory of Higher Categories. I wouldn't advise going all the way to the study of $\infty$-categories, because I think you'll lose the intuition for what makes 2-category theory hard and interesting.

It sounds like the new book from Johnson & Yau "2-Dimensional Categories" might be what you're looking for: https://arxiv.org/abs/2002.06055

While this is not a classical, or perhaps that well-known reference, it does seem to be a comprehensive and detailed take on the subject