2026/2027
Введение в теорию категорий и гомологическую алгебру
ID 1249698
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
2 модуль
Охват аудитории:
для своего кампуса
Преподаватели:
Меркулов Сергей Алексеевич
Язык:
английский
Кредиты:
6
Контактные часы:
72
Course Syllabus
Abstract
The language of categories and functors is one of the most important tools for expressing
universal properties of various mathematical structures as well as for understanding deep interrelations between different areas of mathematics such as, for example, algebra, geometry and topology. After describing the general notions and constructions of the theory of category, such as, for example, limits and colimits constructions, we focus on the symmetric monoidal categories and monoidal functors which we illustrate with some important examples (e.g. as singular chains functor, homology functor, etc). We explain the idea of operad, a mathematical structure which has a particularly nice behavior under monoidal functors and which is used nowadays to transform results obtained in one (say, geometric) category to another (say, algebraic category) and vice versa. The category of chain and cochain complexes is described in detail. The theory of spectral sequences gives us one of the most effective tools to compute (co)homology of concrete complexes. We illustrate these tools with examples from topology and the theory of graph complexes.
Learning Objectives
- The lectures will introduce basic theory and examples, while examples and applications will be explored more deeply in the seminar.
Expected Learning Outcomes
- Fluency in functorial arguments in homological algebra and topology. Familiarity with fundamental examples and calculations.
Course Contents
- Basics of category theory
- Examples and applications
- Basics of homological algebra
- Differential graded algebra
- Categorical examples
Interim Assessment
- 2026/2027 3rd module0,6 * written examination (at the end of 3rd module) + 0,3 homework project + 0,1 in-class tutorials
