The Geometry of Relations

The classical way to study a finite poset (X, ≤ ) using topology is by means of the simplicial complex Δ _X 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 K _X and L _X associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the order complex Δ _X . We give a complete characterization of the simplicial complexes that are the K or L-complexes of some finite poset and prove that K _X and L _X are topologically equivalent to the smaller complexes K′ _X , L′ _X induced by the relation <. More precisely, we prove that K _X (resp. L _X ) 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.