This paper describes the homology of various simplicial complexes associated to set families from combinatorial number theory, including primitive sets, pairwise coprime sets, product-free sets, and coprime-free sets. We present a condition on a set family that results in easy computation of the homology groups, and show that the first three examples, among many others, admit such a structure. We then extend our techniques to address the complexes associated to coprime-free sets and a generalization of primitive sets.