Reference for $(\infty,1)$-Categories

Question

I am looking for an organized source from which I can learn about $(\infty,1)$-categories. I am unable to learn the concept from the $n$lab alone. Here it is said that Lurie called $(\infty,1)$-categories just $\infty$-categories, which makes me wonder whether this article by him addresses what the $n$lab calls $(\infty,1)$-categories.

Ideally, the source would rely on as little prior knowledge as possible, in particular from the realm of enriched categories. Organized definitions are crucial.

Answer 1

In effect, the phrase "$(\infty, 1)$-category" is a cover term for a family of related concepts which are very closely related.

• Quasicategories are certain special simplicial sets. The theory has been extensively developed by Joyal and Lurie, and [Higher topos theory] covers a lot. As you note, Lurie simply calls these "$\infty$-categories".
• Simplicially enriched categories (or simplicial categories for short) are categories equipped with some extra structure. They are concrete and easy to understand conceptually, but not so easy to work with in practice. Unfortunately, there is no handy reference for simplicially enriched categories.
• Complete Segal spaces are certain special bisimplicial sets. The definition (due to Rezk) is very elegant, at least once you understand it. Out of all the notions I mention here, this one has the best "chance" of being internalised in homotopy type theory.
• Segal categories are a kind of hybrid between simplicially enriched categories and complete Segal spaces. The general theory (covering $(\infty, n)$-categories for $0 < n < \infty$) is developed in [Homotopy theory of higher categories].

In every case you will need to know some basic algebraic topology – at the very least, the notions of homotopy, weak homotopy equivalence, and CW complex – and it is very useful to also learn some enriched category theory first. Of course, if you want to understand actual examples of $(\infty, 1)$-categories, you will need to know a bit more algebraic topology or homological algebra.

Answer 2

I can recommend the following introduction:

Moritz Groth, A short course on infinity-categories, http://arxiv.org/abs/1007.2925

Answer 3

A good introductory reference explaining the relationship between the four models mentioned by Zhen Lin is the survey paper

• Julia Bergner, A survey of (infinity,1)-categories, in J. Baez and J. P. May, Towards Higher Categories, IMA Volumes in Mathematics and Its Applications, Springer, 2010, 69-83, pdf.

A good idea would be first to study enriched category theory, as suggested by Zhen Lin, as well as the theory of model categories (e.g. Quillen or Hovey). Then it will be possible to make sense of all the four models mentioned in the survey. I would start by studying simplicial categories, and then moving on to one of the other models. Depending on your applications, quasi-categories are probably the most important model; for that, see e.g. Cisinski's notes here or Joyal's notes here.

Alternatively you could read Emily Riehl's book, available here, which gives a coherent account of everything above, from enriched categories and model categories to quasi-categories.