Categorifications of the 4-term relations via infinitesimal 2-braidings

I will report on recent work with Lucio Cirio on the categorification of the 4-term relations, within the differential geometric context of 2-connections on 2-bundles and their 2-dimensional parallel transport. This will lead to the construction of a Lie-2-algebra of horizontal 2-chord diagrams, as well as a notion of infinitesimal 2-braidings in a Linear 2-category, the latter being closely associated with quasi-invariant tensors in a differential crossed module. We explicitly define a quasi-invariant tensor living in the string Lie-2-algebra.

References: arXiv:1106.0042; arXiv:1207.1132; arXiv:1309.4070.