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We construct a flat (and fake-flat) 2-connection in the configuration spaceof $n$ indistinguishable particles in the complex plane, which categorifies the$sl(2,C)$-Knizhnik-Zamolodchikov connection obtained from the adjointrepresentation of $sl(2,C)$. This will be done by considering the adjointcategorical representation of the string Lie 2-algebra and the notion of aninfinitesimal 2-Yang-Baxter operator in a differential crossed module.Specifically, we find an infinitesimal 2-Yang-Baxter operator in the string Lie2-algebra, proving that any (strict) categorical representation of the stringLie-2-algebra, in a chain-complex of vector spaces, yields a flat and (fakeflat) 2-connection in the configuration space, categorifying the$sl(2,C)$-Knizhnik-Zamolodchikov connection. We will give very detailedexplanation of all concepts involved, in particular discussing the relevanttheory of 2-connections and their two dimensional holonomy, in the specificcase of 2-groups derived from chain complexes of vector spaces.

Categorifying the $sl(2,C)$ Knizhnik-Zamolodchikov Connection via an Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra