The geometry of relations

Details The geometry of relations The geometry of relations

2007-02-07

arxiv.org/abs/math/0702197v1

Mathematics

Combinatorics

10 pages

Scientific paper

There is a canonical way to associate two simplicial complexes K, L to any relation $R\subset X\times Y$. Moreover, the geometric realizations of K and L are homotopy equivalent. This was studied in the fifties by C.H. Dowker. In this article we prove a Galois-type correspondence for relations $R\subset X\times Y$ when X is fixed and use these constructions to investigate finite posets (or equivalently, finite topological spaces) from a geometrical point of view. Given any poset $(X, \leq)$, we define the simplicial complexes K, L associated to the relation $\leq$. In many cases these polyhedra have the same homotopy type as the standard simplicial complex C 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').

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