Zoran Skoda abstract Prague 2024

Braiding phenomena in cyclic homology

Unlike the case of Hochschild (co)homology and most other cohomology theories, the original constructions of cyclic (co)homology in 1980s did not have coefficients. In a variant, Hopf cyclic homology, the coefficients were discovered around 2000 as stable anti-Yetter–Drinfeld modules, reminding of Yetter–Drinfeld (YD) modules familiar from the center construction. Bressler observed that cyclic nerve of a groupoid is determined by the ordinary nerve of its inertia groupoid. Thus I conjectured in late 2002 that passing to the appropriate monoidal category of sheaves one could replace the (sheaves over) inertia by taking the monoidal center of the (sheaves over) original groupoid. This has been proved in 2004 in two variants, well known by Hinich on orbifold case and another by me. As monoidal center involves braiding, it pointed that requiring or adding braidings can provide examples of cyclic homology with coefficients; I constructed some toy examples using standard resolutions and Böhm, Stefan and others independently much more realistic examples using distributive laws. It fits also with work of Kaledin who introduced new kind of traces to treat coefficients. Kowalzig has recently also explained anti- for YD modules in the case of centers of certain bimodule categories. This talk is to outline motivations and my present picture of these braiding (or Yang–Baxter) phenomena in cyclic homology.