We consider a variant of the Seiberg–Witten equations for multiple spinors defined on a closed smooth 4–manifold. Multiple spinors can be thought of as E–valued spinors, where E is some SU(n)–bundle over the base manifold (n > 1). A generalization of the classical Seiberg–Witten equations has been considered by Taubes, Walpuski, Doan, and Zhang, etc., in dimension 3 and dimension 4. Our variant of the equations differs from theirs in the way we define our moment map μ to always be proper. With the properness of the moment map, the technique of Bauer–Furuta in associating a functional of a system of non-linear elliptic PDEs to a certain stable (co)homotopy class of spheres carries over naturally. Such a (co)homotopy class is an invariant of the underlying manifold. This is a work in progress.