# Are there any modern, introductory references detailing the theory of “low dimensional” category theory?

This is related to one of my other open questions. But more generally than my question there, I've found it difficult to teach myself "low dimensional" (i.e. 2 or 3-dimensional) category theory. The n-cat lab is helpful in this regard, but often there is not enough detail presented in the articles to really allow the reader to grasp the concept (beyond just the basic definitions).

Specifically what I'm looking for is a book (or perhaps some lecture notes, or some papers) that detail how one might generalize the usual theorems about the 2-category Cat (i.e. adjoint functor theorem, Yoneda lemma, etc...) to arbitrary 2-categories. In other words, I am looking for a detailed exposition of what is known about the 3-category of 2-categories.

John Gray's book "Adjointness for 2-Categories" is probably the closest thing to this that I can think of, but according to the n-cat lab, the content of the book is rather out-dated.

So are there any similar references to "Adjointness for 2-Categories" which are more up to date, or am I best served just by reading it and translating Gray's terminology into the more modern terminology in my head?

Also, I suppose I should mention: Given a lack of materials specifically for low-dimensional cateogry theory, I would also be satisfied with a reference for the more general $n$-categorical notions, provided that a similar theory (i.e. for the $n+1$ category of $n$-categories), detailing the generalized Yoneda lemma, adjoint functor theorem, etc... is developed.

• I think the modern point of view is that one should learn $(\infty, 1)$ categories before $n$-categories (meaning $(n,n)$ categories), and if you're doing that you might as well do $(\infty, n)$-categories instead. The reason is $(\infty, 0)$ case settles the issue of coherence by reducing it to weak homotopy equivalence, at which point you really can use the idea of $(\infty, n)$ categories being categories enriched over $(\infty, n-1)$-categories. – Hurkyl Sep 28 '17 at 3:58
• @Hurkyl Are there any good references on $(\infty,n)$ categories that you could recommend then? Or, perhaps on $(\infty,1)$-enriched category theory? – Nathan BeDell Sep 28 '17 at 16:52
• @sintrastes There aren't any reasonably introductory references on those subjects so far. – Kevin Carlson Sep 30 '17 at 16:36

Some part of $n$-category theory is just enriched category theory, and you should certainly look at Kelly's book on that if you haven't. But most of the subject deals with weaker notions. There is a significant literature dealing directly with 2-category theory proper. There is no textbook on this material, at least so far, but there is an excellent expository paper by Steve Lack, "A 2-Categories Companion." That was enough, for me, to get into the primary literature, much of which you can find referenced in Lack.
There is relatively little written down for aspects of the cases between $2$ and $\infty$ which aren't just enriched category theory, unfortunately. There is a coherence theorem for tricategories which says that one can't get away with the strict case as one can with 2-categories; in general the naive approach to weakened structures rapidly becomes intractable, and the homotopical approach is the only robust one currently available.