The Geometry of Relations

There is a canonical way to associate two simplicial complexes K, L to any relation R ⊂ X×Y . Moreover, the geometric realizations of K and L are homotopy equivalent. This was studied in the fifties by C.H. Dowker [8]. In this article we prove a Galois-type correspondence for relations R ⊂ X × Y when X is fixed and use these constructions to investigate finite posets (or equivalently, finite T0-spaces) from a geometrical point of view. Given any poset (X,≤), we define the simplicial complexes K, L associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the standard simplicial complex CX of nonempty finite chains in 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 and L are geometrically equivalent to the smaller complexes K , L induced by the relation <. More precisely, we prove that K (resp. L) simplicially collapses to K ′ (resp. L). 2000 Mathematics Subject Classification. 06A06, 06A11, 06A15, 55U05, 55U10, 57Q10.