The Topological Structures of the Orders of Hypergraphs

We provide first a categorical exploration of, and then completion of the mapping of the relationships among, three fundamental perspectives on binary relations: as the incidence matrices of hypergraphs, as the formal contexts of concept lattices, and as specifying topological cosheaves of simplicial (Dowker) complexes on simplicial (Dowker) complexes. We provide an integrative, functorial framework combining previously known with three new results: 1) given a binary relation, there are order isomorphisms among the bounded edge order of the intersection complexes of its dual hypergraphs and its concept lattice; 2) the concept lattice of a context is an isomorphism invariant of the Dowker cosheaf (of abstract simplicial complexes) of that context; and 3) a novel Dowker cosheaf (of chain complexes) of a relation is an isomorphism invariant of the concept lattice of the context that generalizes Dowker's original homological result. We illustrate these concepts throughout with a running example, and demonstrate relationships to past results.