The goal of this course is to expose you to categorical thinking and the general idea of categorification. I will not assume you know about categories already and will tell you what’s necessary to know without getting lost too much in abstract nonsense. I believe that familiarity with basic algebraic structures should be sufficient—but the more you have seen the better it will be. There is an excellent recent book on tensor categories by Etingof, Gelaki, Nikshych, and Ostrik. In the first part of the course we'll discuss the categorical basics (I have typed up lecture notes for this, v0.1 for this course) and in the second part we'll jump into the book. Note that this is just a 2hr course, so we'll only scratch the surface.
The course is virtual, managed through OLAT.
Schedule
| Week | Dates | Sections | Contents |
|---|---|---|---|
| 1 | 26.10.-01.11. | 1.1–1.5 | Categories |
| 2 | 02.11.-08.11. | 2.1–2.4 | Functors |
| 3 | 09.11.-15.11. | 2.5 | Adjunctions |
| 4 | 16.11.-22.11. | 3.1–3.1.19 | Additive categories |
| 5 | 23.11.-29.11. | Rest of 3.1 | Additive categories |
| 6 | 30.11.-06.12. | 3.2 | Abelian categories |
| 7 | 07.12.-13.12. | 3.2.6–3.2.8 + 3.2.22–end of 3.2 + 3.3.1–3.3.10 | Abelian categories (expanded), finite categories |
| 8 | 14.12.-20.12. | 3.4–3.5 + 4.0 | Semisimple categories, Grothendieck groups |
| 9 | 04.01.-10.01. | EGNO Preface + 2.1–2.2 | Monoidal categories |
| 10 | 11.01.-17.01. | EGNO 2.3–2.4 | Examples, monoidal functors |
| 11 | 18.01.-24.01. | EGNO 2.5–2.6 | Examples, graded vector spaces |
| 12 | 25.01.-31.01. | EGNO 2.8–2.9 | MacLane strictness and coherence |
| 13 | 01.02.-07.02. | EGNO 2.10 | Rigidity |
| 14 | 08.02.-14.02. | EGNO 4.1 + 4.2–4.2.7 + 4.5–4.5.2 + 4.9–4.10 | Tensor categories and their Grothendieck rings, categorification |
References
- P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. Tensor categories. Vol. 205. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015.