David Roberts
the manifold of 2-arrows

The manifold of 2-arrows of the Lie fundamental bigroupoid will be locally diffeomorphic to $X^I{\times }_{X\times X}{X}^I$, where $X^I$ is the manifold of 1-arrows, and in should ideally be a covering space (in $\mathrm{Man}$ - Frechet manifolds) of the latter manifold