Functorial characterizations of Pontryagin duality

Author:
David W. Roeder

Journal:
Trans. Amer. Math. Soc. **154** (1971), 151-175

MSC:
Primary 22.20; Secondary 18.00

DOI:
https://doi.org/10.1090/S0002-9947-1971-0279233-X

MathSciNet review:
0279233

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Abstract: Let be the category of locally compact abelian groups, with continuous homomorphisms as morphisms. Let 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 is naturally equivalent to the identity functor in . We characterize by giving necessary and sufficient conditions for an arbitrary contravariant functor to be naturally equivalent to . 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 differs from a natural transformation in that the connecting maps are not required to be morphisms in . We study and classify pseudo-natural transformations in and use this to prove that (*R* denotes the real numbers) is naturally equivalent to if and only if the following three statements are all true:

(1) is isomorphic to *R*,

(2) takes short proper exact sequences to short proper exact sequences, and

(3) takes inductive limits of discrete groups to projective limits and takes projective limits of compact groups to inductive limits.

From this we prove that is naturally equivalent to if and only if is a category equivalence.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0279233-X

Keywords:
Pontryagin duality,
locally compact abelian group,
character group,
compact open topology,
additive category,
functor,
natural transformation,
natural equivalence,
direct sum,
inductive limit,
projective limit,
proper exact sequence,
compactly generated group,
group without small subgroups

Article copyright:
© Copyright 1971
American Mathematical Society