  The groupoid of projectivities, introduced by M. Joswig, serves as a basis
for a construction of parallel transport of graph and more general
$Hom$-complexes. In this framework we develop a general conceptual approach to
the Lovasz Hom-conjecture, recently resolved by E. Babson and D. Kozlov, and
extend their result from graphs to simplicial complexes. The paper also
provides new evidence that the language and methods of groupoids, after being
successfully tested in other major mathematical fields, offer new insights and
perspectives for combinatorial applications.
