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(joint with Fei Ren, Wuppertal University)

Number theory part: I will give a little introduction to K-theoretic Artin maps à la Clausen and K-theoretic Poitou-Tate duality à la Blumberg-Mandell. That's a somewhat new viewpoint on class field theory.

Topology part: LCA groups show up as a surprising model for the compactly supported side of said duality, leading to Clausen's cool way to uniformly describe the non-Galois side of class field theory as K_1(LCA_F) for F the finite/local/global field. As LCA groups do not form an abelian category, one may prefer to work with the (slightly bigger) heart of a t-structure on D^b(LCA). However, this adds new objects (similar to how condensed maths would, but much fewer). These are complexes in D^b up to quasi-iso, and have no reason to be interpretable as a single topological abelian group themselves, right? Why would they? But they do, more or less: With F. Ren we found a (silly? meaningful?) idea to prove an equivalence to a localization of a certain category of (no longer locally compact) topological abelian groups.

(for L2-cohomology fans: There are similarities to how Farber in the 90s extended the category of topological von Neumann G-modules in the context of L2-cohomology to an abelian category, back then called "extended category", but also really just the heart of a t-structure)