Short, new proofs of Dowker duality

TL;DR

This paper introduces three concise proofs of Dowker duality utilizing poset fiber lemmas and defines relational join and product complexes to generalize the duality framework.

Contribution

It provides novel, simplified proofs of Dowker duality and introduces relational complexes that extend the applicability of the duality to broader contexts.

Findings

Established three new proofs of Dowker duality.

Defined relational join and product complexes for face posets.

Connected homologies of complexes and relational complexes via long exact sequence.

Abstract

This paper presents three short, new proofs of Dowker duality using various poset fiber lemmas. We introduce modifications of joins and products of simplicial complexes called relational join and relational product complexes. These relational complexes can be constructed whenever there is a relation between the face posets of simplicial complexes, which includes the context of Dowker duality and covers of simplicial complexes. In this more general setting, we show that the homologies of the simplicial complexes and the relational complexes fit together in a long exact sequence.