Parallel transport of $Hom$-complexes and the Lovasz conjecture

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.