Cosheaf Representations of Relations and Dowker Complexes

TL;DR

This paper introduces a cosheaf-based framework to represent relations via Dowker complexes, making the relation construction a faithful invariant and revealing dualities between the complexes.

Contribution

It develops a cosheaf enrichment of Dowker complexes that captures relations faithfully and establishes a duality between the two complexes.

Findings

Cosheaf representation provides a complete isomorphism invariant for relations.

The construction makes the relation functor faithful.

A duality functor exchanges the two Dowker complexes.

Abstract

The Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction -- vertices for the complex are either the rows or the columns of the matrix representing the relation -- the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes. Finally, we explore a different cosheaf that detects the failure of the Dowker complex itself to be a faithful functor.