The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module

We define the thin fundamental Gray 3-groupoid S-3(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S-3(M) -> C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3-groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables. (C) 2010 Elsevier B.V. All rights reserved.