  We collect evidence that the notion of path-ordered non-abelian integration
admits an extension to two dimensions. We propose the corresponding notion of
non-abelian 2-form along the lines of Lie algebroid theory and argue it is an
appropriate one. The processes of parallel transport and integration turn out
to be subtly different in the 2-dimensional case; we discuss parallel transport
along surfaces and present an indirect definition of a non-abelian integral.
This integral includes, for specific choices of 2-forms, both abelian integrals
and the continuous limit of Baker-Campbell-Hausdorff formula as special cases;
it interpolates between those cases and broadly generalizes them, allowing, for
example, an analog of path-exponential with local, point-depending commutators
to be spoken about. We comment on all these objects, their relations, gauge
symmetries and geometrical meaning, and roughly sketch a plausible
order-by-order procedure for obtaining formulas for non-abelian integrals. The
exposition is reasonably concrete, relying on no notions more abstract than
sections of vector bundles and homotopies.
