A cubical approach to homology theories for hypergraphs

We introduce three homology theories for hypergraphs, namely $\mathrm{\Gamma }$-homology, $\mathrm{\square }$-homology, and $\times$-homology, and show that they are pairwise non-isomorphic and distinct from the embedded homology of hypergraphs. We further introduce a notion of homotopy for hypergraphs that extends the discrete homotopy theory of graphs. Among the homology theories considered, we prove that $\mathrm{\square }$-homology is invariant under this homotopy, whereas $\mathrm{\Gamma }$-homology and $\times$-homology fail to satisfy homotopy invariance. Based on excision, we also identify a distinctive structural behavior exhibited by $\mathrm{\Gamma }$-homology that further differentiates it from $\mathrm{\square }$-homology.