On Duality of Topological Abelian Groups

Let G denote the full subcategory of topological abelian groups consisting of the groups that can be embedded algebraically and topologically into a product of locally compact abelian groups. We show that there is a full coreflective subcategory S of G that contains all locally compact groups and is *-autonomous. This means that for all G,H in S there is an “internal hom” G −◦H whose underlying abelian group is Hom(G,H) and that that makes S into a closed category with a tensor product whose underlying abelian group is a quotient of the algebraic tensor product. Moreover a perfect duality results if we let T denote the circle group and define G∗ = G −◦T. This is essentially a new exposition of work originally done jointly with H. Kleisli [Theory Appl. Categories, 8, 54–62].