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