On two-Dimensional Holonomy

We define the thin fundamental categorical group ${\mathcal{P}}_2(M,\ast )$ of a based smooth manifold $(M,\ast )$ 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 $\text{C}(\text{Gc})$ be a Lie categorical group coming from a Lie crossed module $\text{Gc}=(\text{d}\text{\hspace{0.17em}⁣}:E\to G,\text{tr})$. We construct categorical holonomies, defined to be smooth morphisms ${\mathcal{P}}_2(M,\ast )\to \text{C}(\text{Gc})$, by using a notion of categorical connections, being a pair $(\text{w},m)$, where $\text{w}$ 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 $(\text{w},m)$ satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.