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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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 $ \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.


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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