Introduction to the language of $\infty$-categories

Abstract

The candidate, during his previous bachelor projects, learnt the basic concepts of Algebraic Topology (fundamental group and covering spaces, singular homology and cohomology, first notions about generalized cohomology theories). In this project we are going to study the language of $\infty$-categories, which is essential to understand the classification of topological field theories achieved by Jacob Lurie. At a first stage the candidate will have to deepen his knowledge about basic tools, in particular about higher homotopy groups and simplicial (co)homology within the categorical framework. Then, he will be introduced to higher category theory and he will complete the project dealing with the language of $\infty$-categories.