The starting point for this book was the mini-workshop on “The Homotopy Theory of Homotopy Theories”, held in Caesarea, Israel, in May 2010, and the lecture notes from the talks given there. I was asked by a number of people if those notes might be turned into a more formal manuscript, and this book is the result. In the end, I have omitted some of the topics that were addressed at that workshop, simply because to have included them properly would have greatly increased the length. The topics included in the last chapter of this book were chosen to be in line with some of the applications that were discussed there.

Since 2010, our understanding of (∞, 1)-categories has only increased, and they are being used in a wide range of applications. There are many directions I could have taken with this book, but I have chosen here to give a balanced treatment of the different models and the comparisons between them. I also look at these structures primarily from the viewpoint of homotopy theory, considering model structures and Quillen equivalences. In particular, I do not go into much detail on the treatment of (∞, 1)-categories as generalizations of categories, nor to the development of standard categorical notions in this new context. Joyal and Lurie have treated this topic extensively, extending many categorical notions into the context of quasi-categories in particular [73, 88]. Much less has been done in other models with weak composition, but some work has been done in complete Segal spaces, for example [76].

A number of choices have been made in the presentation given here. While the goal is to give a thorough treatment of the different models, experts on the subject know that just about every model has some technicalities which are intuitively sensible but exceptionally messy to prove. In many cases, I have chosen to omit these technical points and simply refer the reader to the original reference. While this decision makes our book less comprehensive, the hope is that it allows the reader to get the big ideas and most of the details of the proofs without getting sidetracked into often unenlightening, if necessary, combinatorial arguments. I have also deliberately suppressed most set-theoretic points, but have tried to point out where they arise.