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 , K n ] R +of based exterior homotopy classes from a ‘classifying space’ K n . 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.