7.5hp, PhD level course, LP2 2018
Examiner: Thierry Coquand
Category theory originated in algebraic topology, but has found use in recent decades as a unifying language in many areas of mathematics as well as computer science. Homological algebra is an area of mathematics, also originating in algebraic topology, that studies homological functors in an abstract algebraic setting applicable to much of modern mathematics; its modern development uses the language of category theory.
This course will review the core concepts of category theory and then develop the basics of homological algebra using category theoretical language. We expect students to already be familiar with the basics of category theory so that the initial review of category theoretic concepts does not take too much time. The main part of the course is devoted to the basics of homological algebra.
The students meet twice a week, with 2 hours per meeting. They start by exploring the book by Awodey and making sure everyone is familiar with the core concepts of category theory as covered by that book. This should take them about 2 weeks.
The rest of the course is devoted to homological algebra, starting with abelian categories, chain complexes, derived functors and proceeding with a selction of topics in homological algebra as determined by the students. The students can decide topic by topic whether to use the book by Weibel or Chapter IX of the book by Aluffi as basis for this part of the course.
It is up to the students to determine a format of the meetings that fits them. Example setups could be:
- The students read together through a portion of material and discuss it. At the end of each meeting, they select some exercises that each of them solves until the next meeting.
- For each meeting, one student takes charge and leads the presentation one portion of the material. Exercises are solved at home or jointly in the meetings.
After the course is completed, students are expected to be able to follow and perform category theoretical arguments and diagram chases as well as be familiar with standard category theoretical notions such as limits, colimits, adjunctions, exponentials, presheaves, and the Yoneda Lemma. They are expected to be familiar with standard techniques of diagram chasing in abelian categories and basic concepts of homological algebra such as chain complexes and derived functors such as Ext and Tor, know about projective and injective resolutions, perhaps know about the point of view of things from the derived category, and further topics that they have chosen to study.
A take-home exam.