  Hypergraphs are mathematical models for many problems in data sciences. In
recent decades, the topological properties of hypergraphs have been studied and
various kinds of (co)homologies have been constructed (cf. [3, 4, 12]). In this
paper, generalising the usual homology of simplicial complexes, we define the
embedded homology of hypergraphs as well as the persistent embedded homology of
sequences of hypergraphs. As a generalisation of the Mayer-Vietoris sequence
for the homology of simplicial complexes, we give a Mayer-Vietoris sequence for
the embedded homology of hypergraphs. Moreover, as applications of the embedded
homology, we study acyclic hypergraphs and construct some indices for the data
analysis of hyper-networks.
