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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

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

Expected Learning Outcomes

  • Fluency in functorial arguments in homological algebra and topology. Familiarity with fundamental examples and calculations.
Course Contents

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.
  • Differential graded algebra
    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

Assessment Elements

  • non-blocking Midterm exam
  • non-blocking Final Exam
  • non-blocking Seminar participation
Interim Assessment

Interim Assessment

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

Bibliography

Recommended Core Bibliography

  • Курс алгебры, Винберг, Э. Б., 2013

Recommended Additional Bibliography

  • Weibel, C. A. (1994). An Introduction to Homological Algebra. Cambridge University Press.