1) Prerequisites
We will assume basic familiarity with the concepts of category theory, such as functors and natural transformations, limits and colimits, and adjoint functors. (See for example Lecture 1 and 4 of the crash course on category theory https://www.rolandvdv.nl/AT18/#linksat.) Some familiarity with algebraic topology (especially simplices and homotopy groups) will be helpful, but not strictly necessary.
2) Aim of the course
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.
3) Rules about Homework/Exam
Every week will feature practice problems which are designed to deepen your understanding, but need not be handed in. Throughout the course there will be a total of 4 hand-in homework sets. The final exam will be oral or written, depending on the number of participants. If the average homework grade is higher than the exam grade, it will count for 25% (unless the exam grade is less than 5.0). Otherwise the exam grade counts for 100%. In case of a retake, the homework grade does not contribute.
4) Lecture notes/Literature
Standard references are:
J. Lurie, "Higher Topos Theory"
J. Lurie, "Kerodon"
D.-C. Cisinski, "Higher Categories and Homotopical Algebra"
M. Land, "Introduction to Infinity-Categories"
There will be also lecture notes for part of the course.
- Docent: Gijs Heuts
- Docent: Lennart Meier