AbstractIt is well known that cohomology with compact supports is not ahomotopy invariant but only a proper homotopy one. However, as theproper category lacks of general categorical properties, a Brown repre-sentability theorem type does not seem reachable. However, by provingsuch a theorem for the so called exterior cohomology in the completeand cocomplete exterior category, we show that the n-th cohomologywith compact supports of a given countable, locally ïŹnite, ïŹnite dimen-sional relative CW-complex (X,R + ) is naturally identiïŹed with the set[X,K n ] R + ofexteriorbasedhomotopyclassesfrom aâclassifyingspaceâK n . We also show that this space has the exterior homotopy type ofthe exterior Eilenberg-MacLane space for Brown-Grossman homotopygroups of type (R â ,n), Rbeing the ïŹxed coeïŹcient ring. Introduction Properhomotopy theoryis designed to studynon compact topological spacesmodulo deformations which respect the behaviour of these spaces at inïŹnity.In this paper we are concerned with the classiïŹcation of cohomology invari-ants of proper homotopy. Among them, classical cohomology with compactsupports and locally ïŹnite cohomology [20, §3] are specially well adaptedfunctors to study locally compact spaces up to proper homotopy.