I would like to learn some basics about tensor categories and my idea was to get this from the recent book Tensor Categories by P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik (Mathematical Surveys and Monographs, 205, American Mathematical Society, 2015). So, let's have a seminar on this.
The seminar is on Fridays 10–12 in Carslaw 353 (with a few exceptions).
I have written up some comments on the book.
Plan
I would like to see and understand as many examples as possible (there are also interesting things hidden in the exercises!). We don't have to discuss every little detail (like "obvious" commutative diagrams, etc.; I think, I also know what an abelian category is) and I'm happy to accept several things as facts as long as it's stated somehow (like the Perron–Frobenius theorem from linear algebra). For me it's more important to get a feeling for the objects and results than to understand details of proofs (there won't be time for this anyways).
Outline
On page xvi in the preface there's a suggestion for a semester-long graduate course. I think it makes sense to follow that. We should start directly at chapter 2 and introduce auxiliary material from chapters 1 and 3 along the way if necessary. The challenge is to get to chapters 7 and 8 as quickly as possible.
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- Chapter 2: Sections 2.1–2.10 (monoidal categories and functors, the MacLane strictness theorem, rigidity).
- Chapter 4: Sections 4.1–4.9 (basic properties of tensor categories, Grothendieck rings, Frobenius–Perron dimensions).
- Chapter 5: Sections 5.1–5.6 (fiber functors and basic examples of Hopf algebras).
- *Chapter 6: Sections 6.1–6.3 (properties of injective and surjective tensor functors). I suggest to skip this chapter.
- Chapter 7: Sections 7.1–7.12 (exact module categories, categorical Morita theory). We add 7.13--7.15 for me.
- Chapter 8: Sections 8.1–8.14 (examples from metric groups, Drinfeld centers, modular categories).
- Chapter 9: Section 9.1–9.9 and 9.12 (absence of deformations, integral fusion categories, symmetric categories). Maybe we add 9.13 for me.
Schedule
| # | Date | Room | Speaker | Planned |
|---|---|---|---|---|
| 1 | Fr, 09.03.2018, 14–16 | 353 | Ulrich Thiel | 2.1–2.5 (11) |
| 2 | Fr, 16.03.2018, 10–12 | 353 | Ulrich Thiel | 2.6–2.10 (12) |
| 3 | Fr, 23.03.2018, 10–12 | 535 | Thomas Gobet | 4.1–4.9 (11) |
| 4 | Wed, 28.03.2018, 15–17 | 535 | Anthony Henderson | 5.1–5.6 (12) |
| 5 | Fr, 06.04.2018, 10–12 | 353 | Ulrich Thiel | 7.1–7.7 (11) |
| 6 | Fr, 13.04.2018, 10–12 | 353 | Andrew Schopieray (UNSW) | 7.8–7.11 (14) |
| 7 | Fr, 20.04.2018, 10–12 | 353 | Joel Gibson | 7.12–7.15 (11) |
| 8 | Fr, 04.05.2018, 10–12 | 353 | Tarig Abdelgadir (UNSW) | 8.1–8.4 (12) |
| 9 | Fr, 11.05.2018, 10–12 | 353 | Michael Ehrig | 8.5–8.9 (10) |
| 10 | Fr, 18.05.2018, 10–12 | 353 | Ulrich Thiel | 8.10–8.14 (13) |
| 11 | Fr, 25.05.2018, 10–12 | 353 | Joel Gibson and Giulian Wiggins | Graphical calculus |
Review
In 11 sessions (about 2 hours each) we managed to get to the end of chapter 8, which is quite good. We have clearly not discussed and proven everything but still quite a lot. I think one can easily spend twice as much time on the same material when discussing everything with great care. Drawing commutative diagrams takes time! The last lecture on graphical calculus was a bit of an "emergency lecture" but it turned out to fit perfectly there.