I will report on a long term joint project with Lucio Cirio and Florian Schaetz on categorifications of the Knizhnik-Zamolodchikov connection via infinitesimal 2-braidings. In particular, I will describe a categorification of the Drinfeld-Kohno Lie algebra of chord diagrams in the realm of a differential crossed module of horizontal 2-chord diagrams. I will also explain how this categorified Lie algebra arises from a linearization (called an infinitesimal braided 2-category) of the axioms defining a braided monoidal 2-category.
This talk is based on:
T Kohno: Higher holonomy of formal homology connections and braid cobordisms. J. Knot Theory Ramifications, 25, 1642007 (2016)
L. S. Cirio and J Faria Martins: Infinitesimal 2-braidings and differential crossed modules. Advances in Mathematics, Volume 277, 4 June 2015, Pages 426-491
L. S. Cirio and J Faria Martins: Categorifying the Knizhnik–Zamolodchikov connection. Differential Geometry and its Applications, Volume 30, Issue 3, June 2012, Pages 238–261.