  We establish a relation between smooth 2-functors defined on the path
2-groupoid of a smooth manifold and differential forms on this manifold. This
relation can be understood as a part of a dictionary between fundamental
notions from category theory and differential geometry. We show that smooth
2-functors appear in several fields, namely as connections on (non-abelian)
gerbes, as curvatures of smooth functors and as critical points in BF theory.
We demonstrate further that our dictionary provides a powerful tool to discuss
the transgression of geometric objects to loop spaces.
