We construct an invariant of closed spin^c 4-manifolds. This invariant is defined using families of Seiberg-Witten equations and formulated as a cohomology class on a certain abstract simplicial complex. We also give examples of 4-manifolds which admit positive scalar curvature metrics and for which this invariant does not vanish. This non-vanishing result of our invariant provides a new class of adjunction-type genus constraints on configurations of embedded surfaces in a 4-manifold whose Seiberg-Witten invariant vanishes.