Welcome to the class on bicategories in noncommutative geometry, which I was offering during the summer term 2022. The class introduces my recent research work on the bicategory of rings and bimodules and how it may be used to build rings that are interesting for noncommutative geometry. You can find the dates when classes took place in the table below, together with links to the recorded lectures. There are also exercise sheets for this class.
This class is based on a chapter in the draft of a book that I am currently working on. Ocasionally, I will update the pdf file of this chapter on this page.
| Date |
Topic and Recording Link |
| 21.04.2022 |
Motivation from noncommutative geometry, Leavitt algebras, group rings, Morita equivalence |
| 25.04.2022 |
Balanced tensor product, Hom, Eilenberg–Watts Theorem |
| 28.04.2022 |
Morita equivalence through bimodules, examples, finitely generated project module, generator |
| 02.05.2022 |
Morita equivalence, proof of criterion and |
|
Bicategories: motivation and data |
| 05.05.2022 |
Bicategories: Coherence conditions, equivalences in bicategories |
| 09.05.2022 |
Morphisms between bicategories, especially to Rings |
| 12.05.2022 |
Morphisms to Rings as graded rings, some examples of morphisms |
| 16.05.2022 |
Transformations between morphisms of bicategories: motivation, definition, case of Rings |
| 19.05.2022 |
Transformations and graded modules, modifications between transformations, lax cones |
| 23.05.2022 |
Lax covariant representations, lax covariance ring |
| 30.05.2022 |
A case when all transformations are strong, actions of groups by Morita equivalences, fgp modules over C(X) |
| 02.06.2022 |
Group action by Morita equivalences, twisted group action, projective representations |
| 09.06.2022 |
Localisation at rings at elements, matrices, and maps of fgp modules |
| 10.06.2022 12–14 |
Covariance rings as Cohn localisations, Leavitt path algebras as covariance rings |
| 17.06.2022 12–14 |
Permutational bimodules over group rings, iterated monodromy group |
| 20.06.2022 |
Self-similar group, iterated monodromy group, Grigorchuk group |
| 23.06.2022 |
Composition of morphisms, isomorphism of bicategories |
| 24.06.2022 12–14 | Representable homomorphisms, composition of transformations |
| 30.06.2022 | Composition of modifications, equivalence of bicategories, bicategories of morphisms |
| 04.07.2022 |
The Yoneda homomorphism, the Yoneda Lemma, part of its proof (no audio signal for about one hour) |
| 07.07.2022 |
Proof of the Yoneda Lemma for bicategories (no audio signal for last 20 minutes) |
| 08.07.2022 12–14 | Further details of the bicategorical Yoneda lemma, application to representations of homomorphisms, cones and lax cones |
| 14.07.2022 |
Limits and colimits as parts of homomorphisms, whiskering transformations with morphisms |
| 18.07.2022 |
Adjunctions between homomorphisms of bicategories, more compositions for transformations |
| 21.07.2022 |
What has been achieved, what's still to be done? |