Homotopy theory naturally associated with Gamma-homology

Determine a homotopy theory for hypergraphs that is naturally associated with -homology, given that the homotopy defined for hypergraphs preserves -homology but not -homology.

Background

The paper introduces a notion of homotopy for hypergraphs extending discrete homotopy theory for graphs. It proves that -homology is invariant under this homotopy, while examples show that -homology and -homology are not homotopy invariant.

The authors therefore identify as unresolved the task of finding a homotopy theory that is naturally compatible with -homology.