Beijing Institute of Mathematical Sciences and Applications

This course begins with a model-independent axiomatic approach to the theory of ∞-categories. Although this perspective is not entirely formal — since a fully rigorous treatment ultimately requires choosing a specific model (for instance, quasicategories) — it provides a fast and conceptually clear introduction to the subject. We will also explain how classical algebraic topology can be viewed through the lens of ∞-groupoids. The second part of the course is devoted to stable ∞-categories. We will introduce the basic definitions and constructions, including the stabilization of an ∞-category. Finally, we will revisit fundamental notions of stable homotopy theory from the viewpoint of the abstract framework developed earlier.