This book provides an introduction to modern homotopy theory through the lens of higher categories after Joyal and Lurie, giving access to methods used at the forefront of research in algebraic topology and algebraic geometry in the twenty-first century. The text starts from scratch - revisiting results from classical homotopy theory such as Serre's long exact sequence, Quillen's theorems A and B, Grothendieck's smooth/proper base change formulas, and the construction of the Kan–Quillen model structure on simplicial sets - and develops an alternative to a significant part of Lurie's definitive reference Higher Topos Theory, with new constructions and proofs, in particular, the Yoneda Lemma and Kan extensions. The strong emphasis on homotopical algebra provides clear insights into classical constructions such as calculus of fractions, homotopy limits and derived functors. For graduate students and researchers from neighbouring fields, this book is a user-friendly guide to advanced tools that the theory provides for application.

‘Category theory is concerned with the organisation and construction of general mathematical structures, while homotopy theory is devoted to the study of abstract shapes associated to geometric forms. This book is a window into the new field of mathematics emerging from the convergence of these two branches of mathematics … It was conjectured a few decades ago that category theory has a natural extension to quasi-categories (also called infinity-category), a notion introduced by Michael Boardman and Reiner Vogt in the early nineteen-seventies … This book widens and deepens the extension with the addition of a new theory of presheaves inspired by type theory and a new theory of localisation; it proposes an extension of homotopical algebra to quasi-categories, offers new applications, and brings important simplification to earlier works. It is an excellent introduction to the subject and may be used for an advanced course.'

André Joyal - Université du Québec à Montréal, Canada

‘Denis-Charles Cisinski offers a masterful introduction to the world of infinity-categories, illustrating the necessary intuition all throughout. A complete and clear exposition of the foundations leads naturally to a full course teaching us how to handle all aspects of homotopical algebra within the theory.'

Carlos Simpson - French National Center for Scientific Research, Université Côte d'Azur, France

‘In recent years the methods of homotopy theory have seen increasingly wide applications in mathematics, and the framework of abstract homotopy theory has been found to be an important lens through which to view many mathematical structures. This book offers a single, self-contained place to learn about the extensive modern facets of abstract homotopy theory. Readers will appreciate Cisinski's thoughtful choice of details and his carefully articulated philosophical point of view. This is an excellent resource for mathematicians experiencing first contact with the subject and for more seasoned researchers in the area.'

Michael Hopkins - Harvard University

'As someone doing research and advising graduate students in a closely-related area, I am happy to see a book like this in the literature. It will help readers to learn this subject, and to gain a deep understanding of the foundational ideas.'

Julie Bergner Source: MAA Reviews