2025/2026
Введение в бесконечность-один-категории
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для своего кампуса
Преподаватели:
Павлов Александр Борисович
Язык:
английский
Кредиты:
3
Контактные часы:
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.
Learning Objectives
- Learn basic of simplicial sets and their homotopy theory
- Learn properties of the functor "nerve"
- Learn elements of the model category theory: lifting properties, small object argument etc
- Learn language of quasicategories
- Understand proof of Joayl's extendion and lifting theorems
- Understand basic properties of the mapping spaces of quasi-categories
- Learn Joyal's model structure on the category of simplicial sets
Expected Learning Outcomes
- To learn basics of model category theory
- To understand examples of model structures
- State definition of a model category.
- Understand the homotopy theory of simplicial sets.
- Know the small object argument.
- Know what are left/right/inner anodyne maps and corresponding fibrations.
- Formulate and know how to apply Joyal's criterion for infnity-categories.
- Formulate and know how to apply Joyal's lifting/extension theorem.
- Apply methods of the infinity category theory to your projects.
Course Contents
- Simplicial sets
- Language of Model Categories
- Small Object Argument
- Language of infinity-categories
- Joins and Slices
- Joyal's Extension and Lifting theorems
- Isofibrations
Bibliography
Recommended Core Bibliography
- A combinatorial introduction to topology, Henle, M., 1994
- An introduction to homological algebra, Rotman, J. J., 2009
- An introduction to the language of category theory, Roman, S., 2017
- Methods of homological algebra, Gelfand, S. I., 2003
Recommended Additional Bibliography
- Allen Hatcher. (2001). Algebraic topology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.5FA4491E
- An introduction to category theory, Krishnan, V. S., 1981
- Hollander, S. (2001). A Homotopy Theory for Stacks. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.math%2f0110247
- James, I. M. Handbook of Algebraic Topology: North Holland: p.1324 , 1995. - ISBN 978-0-444-81779-2