The Rectangle Complex of a Relation

TL;DR

This paper introduces the rectangle complex of a relation, demonstrating its homotopy equivalence to the Dowker complex, providing a conceptual proof of functorial Dowker's Theorem in topological data analysis.

Contribution

It constructs the rectangle complex and proves its homotopy equivalence to the Dowker complex, offering a new, conceptual proof of functorial Dowker's Theorem.

Findings

Rectangle complex is homotopy equivalent to Dowker complex

Provides a short, conceptual proof of functorial Dowker's Theorem

Enhances understanding of topological data analysis tools

Abstract

We construct a simplicial complex, the rectangle complex of a relation R, and show that it is homotopy equivalent to the Dowker complex of R. This results in a short and conceptual proof of functorial versions of Dowker's Theorem used in topological data analysis.