Denis-Charles Cisinski is Professor of Mathematics at the Universität Regensburg, Germany. His research focuses on homotopical algebra, category theory, K-theory and the cohomology of schemes. He is also the author of a monograph entitled Les préfaisceaux comme modèles des types d'homotopie (2007).

'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

• Preface; 1. Prelude; 2. Basic homotopical algebra; 3. The homotopy theory of ∞-categories; 4. Presheaves: externally; 5. Presheaves: internally; 6. Adjoints, limits and Kan extensions; 7. Homotopical algebra; References; Notation; Index.