Hom Complexes of Set Systems

A set system is a pair S = (V (S),∆(S)), where ∆(S) is a family of subsets of the set V (S). We refer to the members of ∆(S) as the stable sets of S. A homomorphism between two set systems S and T is a map f : V (S) → V (T ) such that the preimage under f of every stable set of T is a stable set of S. Inspired by a recent generalization due to Engström of Lovász’s Hom complex construction, the author associates a cell complex Hom(S, T ) to any two finite set systems S and T . The main goal of the paper is to examine basic topological and homological properties of this cell complex for various pairs of set systems.