A category consists of a collection of objects, a collection of morphisms, and a rule for how to compose such morphisms. This notion is ubiquitous in mathematics; any kind of objects one wishes to study (sets, vector spaces, manifolds, schemes, etc.) can usually be organized into a category. A higher category (or ∞-category) has objects and morphisms between objects, but also 2-morphisms between morphisms, 3-morphisms between 2- morphisms, etc. This notion was first invented in the context of homotopy theory, where morphisms are continuous maps, 2-morphisms are homotopies between them, and so on. Over the past 20 years, higher category theory has seen enormous development through the works of Joyal and Lurie and has rapidly found many applications to other fields, such as algebraic geometry, number theory, and representation theory. It is now a fundamental tool in modern mathematical research.

The aim of this course is to develop the basic notions of the theory of ∞-categories. A student who completes the course should then be able to dive into any of the standard references on the subject and read research papers that depend on the methods of higher category theory. Some of the topics we will cover are the following:

• Simplicial sets
• Nerves of categories
• Infinity-groupoids and ∞-categories
• Limits and colimits in ∞-categories
• Functors between ∞-categories
• Fibrations of ∞-categories

This is an advanced course, intended to introduce students to a set of methods very important to modern research. We will assume basic familiarity with the concepts of category theory, such as functors and natural transformations, limits and colimits, and adjoint functors. Some familiarity with algebraic topology will be helpful, but not strictly necessary.