Functorial characterizations of Pontryagin duality

Abstract

Let Y 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 Y its character group and to each morphism its adjoint morphism. The Pontryagin duality theorem is then the statement that x a x is naturally equivalent to the identity functor in Y. We characterize x by giving necessary and sufficient conditions for an arbitrary contravariant functor p: Y -Y 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 Y differs from a natural transformation in that the connecting maps are not required to be morphisms in Y. We study and classify pseudo-natural transformations in Y and use this to prove that (R denotes the real numbers) q is naturally equivalent to x if and only if the following three statements are all true: (1) F(R) is isomorphic to R, (2) q takes short proper exact sequences to short proper exact sequences, and (3) p takes inductive limits of discrete groups to projective limits and takes projective limits of compact groups to inductive limits. From this we prove that p is naturally equivalent to x if and only if q' is a category equivalence. Pontryagin duality deals with the relationship between a locally compact abelian group G and its character group G. This correspondence G -H G^ extends to a contravariant functor, which we denote by X, from the category Y of locally compact abelian groups to itself. In this paper we give three separate characterizations of X Presented in part to the Society, January 23, 1969, under the title Two characterizations of Pontryagin duality; received by the editors September 15, 1969 and, in revised form, March 31, 1970. AMS 1970 subject classifications. Primary 22B05, 43A40, 43A95, 18-XX, 18B99; Secondary 18A20, 18A30, 18E05.