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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.

Thank you in advance

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    $\begingroup$ What aspect of 2-category theory is troubling you specifically? Is it the theory of 2-categorical or bicategorical limits? $\endgroup$
    – Todd Trimble
    Commented Jan 4, 2015 at 14:11
  • $\begingroup$ 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? $\endgroup$
    – Fernando
    Commented Jan 4, 2015 at 14:36
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    $\begingroup$ Have you looked at the nLab? This might get you started: ncatlab.org/nlab/search?query=kan+extension $\endgroup$
    – Todd Trimble
    Commented Jan 4, 2015 at 14:56
  • $\begingroup$ 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 $\endgroup$
    – Fernando
    Commented Jan 4, 2015 at 15:13

3 Answers 3

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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

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  • $\begingroup$ I think a nice addition to this list would be „Pullbacks equivalent to pseudopullbacks“ by Joyal and Street. It's a short article (4 pages) stating that strict pullbacks along isofibrations in Cat coincide with pseudo bipullbacks. This generalizes to arbitrary pullbacks and pushouts along 1-cells which are (co)representably isofibrations (i.e. all components of their representable transformation B(-,f) are isofibrations). The corepresentably isofibrant functors in Cat are identified as exactly those injective on objects. $\endgroup$
    – Julia Path
    Commented Jun 6, 2022 at 13:41
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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.

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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

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    $\begingroup$ I have the book and read some parts of it. It is mostly rather technical and in many places very surface level. For example the chapter on limits defines lax, pseudo and strict limits as well as their bilimit variants, but thats kind of it. It doesn't give any examples on them and, it doesn't discuss weighted limits and as a result misses the relations between lax, pseudo and strict limits. The chapter also gives no examples of limits at all. No inserters or equifiers or anything like that is mentioned. Not even in exercises. $\endgroup$
    – Julia Path
    Commented Jun 6, 2022 at 13:19
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    $\begingroup$ I think the book may be good as a reference on the technical foundations of the theory as well as a technical introduction to the subject, but at least for understanding limits I found the chapter of relatively little use. Articles like the ones mentioned above were a lot more useful. $\endgroup$
    – Julia Path
    Commented Jun 6, 2022 at 13:24

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