Brown representability for exterior cohomology and cohomology with compact supports

It is well known that cohomology with compact supports is a proper homotopy invariant. However, as the proper category lacks general categorical properties, a Brown representability theorem type does not seem reachable. By proving such a theorem for the so‐called exterior cohomology in the complete and cocomplete exterior category, we show that the n th cohomology with compact supports of a given countable, locally finite, finite‐dimensional relative CW‐complex (X,R+) is naturally identified with the set [X,Kn]R+ of based exterior homotopy classes from a ‘classifying space’ Kn. We also show that this space has the exterior homotopy type of the exterior Eilenberg–MacLane space for Brown–Grossman homotopy groups of type (R∞,n), R being the fixed coefficient ring.