Functorial characterizations of Pontryagin duality

Abstract:

Let $\mathcal {L}$ be the category of locally compact abelian groups, with continuous homomorphisms as morphisms. Let $\chi :\mathcal {L} \to \mathcal {L}$ denote the contravariant functor which assigns to each object in $\mathcal {L}$ its character group and to each morphism its adjoint morphism. The Pontryagin duality theorem is then the statement that $\chi \circ \chi$ is naturally equivalent to the identity functor in $\mathcal {L}$. We characterize $\chi$ by giving necessary and sufficient conditions for an arbitrary contravariant functor $\varphi :\mathcal {L} \to \mathcal {L}$ to be naturally equivalent to $\chi$. 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 $\mathcal {L}$ differs from a natural transformation in that the connecting maps are not required to be morphisms in $\mathcal {L}$. We study and classify pseudo-natural transformations in $\mathcal {L}$ and use this to prove that (R denotes the real numbers) $\varphi$ is naturally equivalent to $\chi$ if and only if the following three statements are all true: (1) $\varphi (R)$ is isomorphic to R, (2) $\varphi$ takes short proper exact sequences to short proper exact sequences, and (3) $\varphi$ takes inductive limits of discrete groups to projective limits and takes projective limits of compact groups to inductive limits. From this we prove that $\varphi$ is naturally equivalent to $\chi$ if and only if $\varphi$ is a category equivalence.