Functorial characterizations of Pontryagin duality

Abstract

Let -S? be the category of locally compact abelian groups, with continuous homomorphisms as morphisms. Let x'--&'-* & denote the contravariant functor which assigns to each object in & its character group and to each morphism its adjoint morphism. The Pontryagin duality theorem is then the statement that X x is naturally equivalent to the identity functor in if. We characterize x by giving necessary and sufficient conditions for an arbitrary contravariant functor q>: & -* -S? to be naturally equivalent to x-A sequence of morphisms is called proper exact if it is exact in the algebraic sense and is composed of morphisms each of which is open considered as a function onto its image. A pseudo-natural transformation between two functors in .S? diners from a natural transformation in that the connecting maps are not required to be morphisms in SC. We study and classify pseudo-natural transformations in S and use this to prove that (R denotes the real numbers) <p is naturally equivalent to x if and only if the following three statements are all true: