An Introduction to the Theory of Categories and Homological Algebra
- The lectures will introduce basic theory and examples, while examples and applications will be explored more deeply in the seminar.
- Fluency in functorial arguments in homological algebra and topology. Familiarity with fundamental examples and calculations.
- Basics of category theoryCategories, functors, natural transformations, adjoints, limits and colimits
- Examples and applicationsA selection of examples and applications from algebraic topology, commutative algebra, representation theory, and sheaf theory.
- Basics of homological algebraHom and tensor of complexes. Exactness. Cones. Long exact sequences. Derived functors. Ext and Tor.
- Differential graded algebraDG modules and more general DG categories
- Categorical examplesFunctors between topological spaces, simplicial sets, and chain complexes. Free and forgetful functors, abelianisation.
- Interim assessment (4 module)0.5 * Final Exam + 0.4 * Midterm exam + 0.1 * Seminar participation