What textbooks on higher category theory are there? What books do you recommend? I am looking for self-contained introductions, no research reports. There are lots of informal summaries and arXiv papers, but I am really only asking for textbooks here.

I know of Lurie's Higher Topos Theory, which "only" treats $(\infty,1)$-categories. I am looking for books which treat $\infty$-categories in general. Then I know of Leinster's Higher Operads, Higher Categories, which is from 2004. Is it still up to date? Is Leinster's book the best introduction to the subject? What do you think of Higher-Dimensional Categories: an illustrated guide book by Cheng and Lauda, which is also from 2004 and still a draft? Is it too informal when one really wants to work with the concepts?

Bonus question: Meanwhile, is there some "preferred" definition of an $\infty$-category among the dozen definitions which have been studied?

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    $\begingroup$ The question is far from being settled. That is why there are no textbooks. $\endgroup$ – Zhen Lin May 10 '15 at 19:47
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    $\begingroup$ There are three textbooks, at least, and my question is in particular if someone has some experience with them. $\endgroup$ – Martin Brandenburg May 11 '15 at 10:20
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    $\begingroup$ Lurie's book is a textbook, but as you say, it not a textbook on $\infty$-categories. Leinster's book is not a textbook on $\infty$-categories in the same sense that CWM is a textbook on categories – you won't even find a higher Yoneda lemma in there. The same goes for the book of Cheng and Lauda. $\endgroup$ – Zhen Lin May 11 '15 at 11:28
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    $\begingroup$ Re: bonus question, a first answer it probably depends on who you ask. Sometimes people even use multiple models in a single paper -- recently I was at a talk where quasi-categories and complete Segal spaces were used (plus a brand new model based on striation sheaves). A second answer is that as far as I've noticed, there are a lot of attempts to try to make every argument model-independent. (Unrelated: there's also Higher Algebra by Lurie, but I don't know what qualifies as a textbook?) $\endgroup$ – Najib Idrissi May 11 '15 at 13:50
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    $\begingroup$ Again I wonder why answers are written as comments ... $\endgroup$ – Martin Brandenburg May 13 '15 at 6:48

1. First of all, make sure to have a few references on category theory available. Good ones include:

2. It also pays of to learn about the insights leading to $\infty$-categories before learning about their theory proper. A good reference here is John Baez's An Introduction to $n$-Categories. Another one is Section 1.2 of Hellstrøm-Finnsen's thesis.

3. $\infty$-Categories require two fundamental prerequisites: model category theory and simplicial sets.

Simplicial Sets. Friedman's An elementary illustrated introduction to simplicial sets is a marvelous introduction for beginners. For more in-depth references, there are May's Simplicial Objects in Algebraic Topology, and Simplicial Homotopy Theory by Goerss–Jardine.

Model Categories. Good references for model category theory include:

4. ($\infty$-Categories, finally) It is hard to capture in a precise way the idea of an $\infty$-category as a set of objects, together with a set of morphisms, a set of $2$-morphisms, and so on. There are two ways of approaching this difficulty, one traditional, the other very recent.

Via Quasicategories. The traditional one is to use models for $\infty$-categories. One such model is given by a special kind of simplicial set called a quasicategory. This is the approach developed by Joyal and Lurie. For learning the theory of quasicategories, there are:

Via $\infty$-Cosmoi. The second one is the model-independent approach of Riehl and Verity (which is currently being developed). Instead of axiomatizing what $\infty$-categories are via models, Riehl–Verity axiomatize the mathematical object in which $\infty$-categories live, and call it an $\infty$-cosmos.

When working with an specific model for $\infty$-categories, one is often lead to complicated arguments involving its combinatorics. On the other hand, in Riehl–Verity's framework, it is possible to prove statements about $\infty$-categories in a much simpler, model-independent, way

Riehl and Verity are currently compiling their work in a textbook, called Elements of $\infty$-Category Theory.

Extra References


Background on model categories and simplicial sets


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