Higher category theory and homotopical algebra

The purpose of this course is to give a proper introduction to higher category theory, into the form used in modern Algebraic Topology and Algebraic Geometry, and to emphasize that Category Theory and Algebraic Topology arguably are two aspects of a single aim. The purpose of the lecturer is to give a basic account which is both rigorous and accessible. While giving an introduction to the theory of quasi-categories, as developed by Joyal and Lurie, we will also give a basic introduction to Quillen's theory of model categories and will give many examples in order to connect with common practice and traditional preoccupations of Algebraic Topology, as well as to illustrate the strength of this theory. However, the Leitfaden of these lectures will be the problem of strictifying categorical structures, which we will consider in many ways: we will look at quite a few versions of the Yoneda embedding, and see how the Joyal model structure exhibits Higher Category Theory as a semantic of the formal language of ordinary category theory. The prerequisite formally are few: a basic knowledge of Algebraic Topology and of Category Theory: notions of adjoint functors, limits and colimits, example of non-trivial functors coming from Algebraic Topology or Homological Algebra (e.g. singular homology), as well as examples of homotopy invariance and of long exact sequences (e.g. the content of the course Algebraic Topology I). A basic knowledge of simplicial methods might help, but it is not mandatory. It should be emphasized that, although these lectures will demand a certain level of abstraction and of mathematical sophistication, we will really try to make them as accessible as possible.