André Joyal (Université du Québec à Montréal)

The theory of quasi-categories and its applications

The goal of the course is to present the theory of quasi-categories and some of its applications. The notion of quasi-category was introduced by Boardman and Vogt in their work on homotopy invariant algebraic structures. Category theory has been extended to quasi-categories. The first lectures will build on the analogies between the two theories.   A description will be given of the model structure for quasi-categories and of the equivalences with other model structures. The following concepts will be introduced: adjoint maps, limits and colimits, the factorisation system of right fibrations and final maps, the factorisation system of quasi-localisation and conservative maps, the contravariant model structure, and the Yoneda lemma. The basic notions of quasi-algebra will also be introduced with examples. In the second part of the course, the differences between category theory and the theory of quasi-categories will be discussed. Applications to homotopy theory, homotopical algebra, higher topos theory, stable homotopy theory, and higher category theory will also be discussed.