Brown representability for exterior cohomology and cohomology with compact supports

Abstract

It is well known that cohomology with compact supports is not a homotopy invariant but only a proper homotopy one. However, as the proper category lacks of general categorical properties, a Brown representability theorem type does not seem reachable. However, 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 $(X,\mathbb{R}_+)$ is naturally identified with the set $[X,K_n]^{\mathbb{R}_+}$ of exterior based 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 $(R^\infty,n)$, $R$ being the fixed coefficient ring.