Dowker’s theorem for higher-order relations

Given a relation R⊆I×J between two sets, Dowker’s Theorem (1952) states that the homology groups of two associated simplicial complexes—now known as Dowker complexes—are isomorphic. In its modern form, the full result asserts a functorial homotopy equivalence between the two Dowker complexes. What can be said about relations defined on three or more sets? We present a simple generalization to ‘multiway’ relations of the form R⊆I1×I2×⋯×Im. The theorem asserts functorial homotopy equivalences between m multiway Dowker complexes and a variant of the rectangle complex of Brun and Salbu from their recent short proof of Dowker’s Theorem. Our proof uses Smale’s homotopy mapping theorem and factors through a ‘cellular Dowker lemma’ that expresses the main idea in more general form. To make the geometry more transparent, we work with a class of spaces called ‘prod-complexes’ then transfer the results to simplicial complexes through a ‘simplexification’ process. We conclude with a detailed study of ternary relations, identifying seven functorially defined homotopy types and twelve natural transformations between them.