2020/2021
An Introduction to the Theory of Categories and Homological Algebra
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
3, 4 module
Instructors:
Christopher Ira Brav
Language:
English
ECTS credits:
6
Contact hours:
72
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 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.
