Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] I. Bicur and A. Deleanu, Introduction to the theory of categories and functors, Wiley, New York, 1968. MR 0236236 (38:4534)
  • [2] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York and Springer-Verlag, Berlin, 1963. MR 28 #158. MR 551496 (81k:43001)
  • [3] K. H. Hofmann, Categories with convergence, exponential functors, and the cohomology of compact abelian groups, Math. Z. 104 (1968), 106-140. MR 37 #4195. MR 0228615 (37:4195)
  • [4] S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1965. MR 33 #5416. MR 0197234 (33:5416)
  • [5] S. Mac Lane, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Academic Press, New York, and Springer-Verlag, Berlin, 1963. MR 28 #122.
  • [6] B. Mitchell, Theory of categories, Pure and Appl. Math., vol. 17, Academic Press, New York, 1965. MR 34 #2647. MR 0202787 (34:2647)
  • [7] M. Moskowitz, Homological algebra in locally compact abelian groups, Trans. Amer. Math. Soc. 127 (1967), 361-404. MR 35 #5861. MR 0215016 (35:5861)
  • [8] L. S. Pontryagin, Topological groups, GITTL, Moscow, 1954; English transl., Gordon and Breach, New York, 1966. MR 17, 171; MR 34 #1439. MR 0201557 (34:1439)
  • [9] K. Suzuki, Notes on the duality theorem of non-commutative topological groups, Tôhoku Math. J. (2) 15 (1963), 182-186. MR 26 #6293. MR 0148788 (26:6293)
  • [10] A. Weil, L'intégration dans les groupes topologiques et ses applications, Hermann, Paris, 1951.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22.20, 18.00

Retrieve articles in all journals with MSC: 22.20, 18.00


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

American Mathematical Society