Title of the course: Towards higher categories
Instructor: Dr. Joseph Helfer
Institution: Simons Center for Geometry and Physics
Dates: 25 August – 7 September 2025
Prerequisites: Some category theory, maybe some algebraic topology/homotopy theory, maybe some algebraic geometry
Level: Graduate, advanced undergraduate
Abstract: The theory of (∞,1)-categories (also known as quasi-categories or ∞-categories) has come to play a prominent role in modern homotopy theory, representation theory, and algebraic geometry. It is also part of the more general theory of higher categories, which is of general interest for the foundations of mathematics.
As this theory requires a fair amount of categorical background which I don’t want to expect or require participants to have, we will instead be studying this background material, in order to motivate the theory of higher categories, and prepare participants for a more in depth study of it.
Possible topics include: presheaf and sheaf categories, model categories, presentable categories, Grothendieck fibrations, stacks, enriched categories, bicategories, strict ω-categories, stable homotopy theory.
We will try to view as many examples as we can along the way.
Language: English
Textbook: https://www.jojhelfer.com/nesin2025/