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Regular version of the site
2020/2021

## An Introduction to the Theory of Categories and Homological Algebra

Type: Optional course (faculty)
When: 3, 4 module
Open to: students of one campus
Language: English
ECTS credits: 6

### Course Syllabus

#### Abstract

The theory of categories provides tools for studying the relations between different areas of mathematics, especially between topology, geometry, and algebra. Homological algebra in particular grew out algebraic topology and is now widely used in representation theory and algebraic geometry.

#### 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
Categories, functors, natural transformations, adjoints, limits and colimits
• Examples and applications
A selection of examples and applications from algebraic topology, commutative algebra, representation theory, and sheaf theory.
• Basics of homological algebra
Hom and tensor of complexes. Exactness. Cones. Long exact sequences. Derived functors. Ext and Tor.
DG modules and more general DG categories
• Categorical examples
Functors between topological spaces, simplicial sets, and chain complexes. Free and forgetful functors, abelianisation.

#### Assessment Elements

• Midterm exam
• Final Exam
• Seminar participation

#### Interim Assessment

• Interim assessment (4 module)
0.5 * Final Exam + 0.4 * Midterm exam + 0.1 * Seminar participation

#### Recommended Core Bibliography

• Курс алгебры, Винберг, Э. Б., 2013