2025/2026

Introduction to Infinity-one-categories

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

When: 1, 2 module

Open to: students of one campus

Instructors: Alexander Pavlov

Language: English

Contact hours: 30

Course Syllabus

Abstract

This is a topic course in category theory and homotopical topology. On one hand an \infty-category is some kind of higher categorical structure, on the other it encodes the data of a homotopy theory. The theory of \infty-categories generalizes homotopy categories of topological spaces (more generally homotopy categories of model categories) and derived categories of abelian categories. Informally speaking, an (\infty, 1)-category is a category enriched in topological spaces. There are several ways (models) to make this definition precise: complete Segal spaces, Segal categories, quasi-categories etc. We will discuss those models and why they are equivalent and develop the “standard toolkit” of the category theory in those models. The theory of \infty-categories has many applications in modern mathematics such as the proof of Weil’s conjecture on Tamagawa numbers over function fields by Lurie and Gaitsgory, or the modern approach to p-adic Hodge theory by Bhatt, Morrow and Scholze, for instance. In the last part of the course we will cover basics of stable (\infty, 1)-categories as modern foundation of homological algebra. This point of view on homological algebra have several advantages that led to it becoming basis of derived algebraic geometry.