The Geometry of Relations

Abstract

The classical way to study a finite poset (X, ≤) using topology is by means of the simplicial complex ΔX of its nonempty chains. There is also an alternative approach, regarding X as a finite topological space. In this article we introduce new constructions for studying X topologically: inspired by a classical paper of Dowker (Ann Math 56:84-95, 1952), we define the simplicial complexes KX and LX associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the order complex ΔX. We give a complete characterization of the simplicial complexes that are the K or L-complexes of some finite poset and prove that KX and LX are topologically equivalent to the smaller complexes K′X, L′X induced by the relation ≤. More precisely, we prove that KX (resp. LX) simplicially collapses to K′X (resp. L′X). The paper concludes with a result that relates the K-complexes of two posets X, Y with closed relations R ⊂ X × Y.