  We develop the theory of orbibundles from a geometrical viewpoint using
diffeology. One of our goals is to present new tools allowing to calculate
invariants of complex hyperbolic disc orbibundles over $2$-orbifolds appearing
in the geometry of $4$-manifolds. These invariants are the Euler number of disc
orbibundles and the Toledo invariant of $\mathrm{PU}(2,1)$-representations of
$2$-orbifold groups.
