Bicategories in noncommutative geometry (Prof. Dr. Ralf Meyer)

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.

Lectures and recording links

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 12–14Representable homomorphisms, composition of transformations
30.06.2022Composition 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 12–14Further 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?

Exercise sheets