On two-dimensional holonomy

We define the thin fundamental categorical group ${\mathcal P}_2(M,*)$ 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 rank-2 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 $\mathcal{C}(\mathcal{G})$ be a Lie categorical group coming from a Lie crossed module ${\mathcal{G}= (\partial\colon E \to G,\triangleright)}$. We construct categorical holonomies, defined to be smooth morphisms ${\mathcal P}_2(M,*) \to \mathcal{C}(\mathcal{G})$, by using a notion of categorical connections, being a pair $(\omega,m)$, where $\omega$ 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 $(\omega,m)$ satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.