2024 Winter Mentoring: Introduction to Category Theory

During this winter vacation, I conducted a 8-lecture course Introduction to Category Theory as a part of the 2024 Winter Mentoring program in the College of Natural Sciences, Seoul National University. The objective of the course was to offer a category theoretic viewpoint on undergraduate mathematics, and to introduce basic concepts of homological algebra. I also created a problem set as a supplementary material for the course.

Syllabus

Course Description

Many branches of mathematics define certain types of objects, and study morphisms between them. Examples include vector spaces and linear transformations for linear algebra, groups and homomorphisms for group theory, and topological spaces and continuous maps for point-set topology. Category theory formalizes these mathematical structures as categories, and study relationships between them through functors, thereby revealing their omni-disciplinary properties and fundamental characteristics.

Undergraduate mathematics students may have encountered glimpses of category theory in their coursework. Have you ever noticed a sense of uniformity across various fields? When do we consider two types of instances to be the same? In what contexts do we use descriptors like universal, natural, and free used? This course seeks to address students’ curiosity about these questions.

The aim of this course is to reexamine undergraduate mathematics from the viewpoint of category theory, thereby enhancing students’ mathematical maturity and categorical insight. In the first half of the course, we introduce concepts of category theory and explore their interconnections. In the second half, we provide a rigorous treatment of homology and cohomology from a categorical perspective. Rather than delving deeply into theory, the focus is on providing various examples encountered in undergraduate mathematics.

While no specific theoretical prerequisites are required, students are encouraged to have taken courses such as Linear Algebra, Modern Algebra, and Introduction to Topology to understand the diverse examples discussed in the class.

Course Meeting Times

2 sessions / week, 2 hours / session

References

• 이인석, 《학부 대수학 강의 I: 선형대수와 군》, 개정판. 서울대학교출판문화원, 2015.
• 이인석, 『Homological Algebra 강의록』, 전자판. 2020. (Online)
• Lang, S., Algebra, Third Edition. Springer, 2002.
• Hungerford, T. W., Algebra, First Edition. Springer, 1974.
• MacLane, S., Categories for the Working Mathematician, Second Edition. Springer, 1978.
• Riehl, E., Category Theory in Context, First Edition. Dover, 2016.
• Weibel, C. A., An Introduction to Homological Algebra, First Edition. Cambridge University Press, 1995.
• MacLane, S., Homology, First Edition. Springer, 1995.
• Northcott, D. G., An Introduction to Homological Algebra, First Edition. Cambridge University Press, 1960.
• Hatcher, A., Algebraic Topology, First Edition. Cambridge University Press, 2001.
• Lee, J., Introduction to Smooth Manifolds, Second Edition. Springer, 2012.
• Lang, S., Topics in cohomology of groups, First Edition. Springer, 1996.

Course Outline

Lecture 1: Category and Universality (2024.1.9.)

• Category
• Categorical Concept
• Universal Object
• Vocabulary: “Universality”

Lecture 2: Functor and Naturality (2024.1.11.)

• Functor
• Natural Transformation
• Vocabulary: “Naturality”
• Isomorphism and Equivalence of Categories

Lecture 3: Adjunction and Freeness (2024.1.16.)

• Adjunction
• Free Functor
• Vocabulary: “Freeness”

Lecture 4: Limit and Colimit (2024.1.18.)

• Kernel & Cokernel
• Inverse Limit & Direct Limit
• Limit & Colimit
• Preservation of Limits under Adjoint Functors

Lecture 5: Classical Homology & Cohomology (2024.1.25.)

• Simplicial Homology
• Singular Homology & Cohomology
• de Rham Cohomology

Lecture 6: Abelian Category (2024.1.30.)

• Abelian Category
• Additive Functor
• Chain Complex & Cochain Complex
• Homology & Cohomology
• Exactness

Lecture 7: Homology Long Exact Sequence (2024.2.1.)

• Snake Lemma
• Homology Long Exact Sequence
• Naturality of Connecting Morphism

Lecture 8: Derived Functor (2024.2.8.)

• Projective & Injective
• Derived Functor
• Examples of Derived Functor

Problem Set

This is the problem set I created as a supplementary material for the course.