  A foundation is laid for a theory of combinatorial groupoids, allowing us to
use concepts like ``holonomy'', ``parallel transport'', ``bundles'',
``combinatorial curvature'' etc. in the context of simplicial (polyhedral)
complexes, posets, graphs, polytopes and other combinatorial objects. A new,
holonomy-type invariant for cubical complexes is introduced, leading to a
combinatorial ``Theorema Egregium'' for cubical complexes non-embeddable into
cubical lattices. Parallel transport of Hom-complexes and maps is used as a
tool for extending Babson-Kozlov-Lovasz graph coloring results to more general
statements about non-degenerate maps (colorings) of simplicial complexes and
graphs.
