On Two-Dimensional Holonomy

We define the thin fundamental categorical group P2(M, ∗) of a based smooth manifold (M, ∗) as the categorical group whose objects are rank-1 homotopy classes of based loops on M , and whose morphisms are rank2 homotopy classes of homotopies between based loops on M . Here two maps are rank-n homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (∂ : E → G, ⊲). We construct categorical holonomies, defined to be smooth morphisms P2(M, ∗) → C(G), by using a notion of categorical connections, being a pair (ω,m), where ω is a connection 1-form on P , a principal G bundle over M , and m is a 2-form on P with values in the Lie algebra of E, with the pair (ω,m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.