Pontryagin Duality - Categorical Considerations

Categorical Considerations

It is useful to regard the dual group functorially. In what follows, LCA is the category of locally compact abelian groups and continuous group homomorphisms. The dual group construction of G^ is a contravariant functor LCA → LCA, represented (in the sense of representable functors) by the circle group T, as G^=Hom(G,T). In particular, the iterated functor G → (G^)^ is covariant.

Theorem. The dual group functor is an equivalence of categories from LCA to LCAop.

Theorem. The iterated dual functor is naturally isomorphic to the identity functor on LCA.

This isomorphism is analogous to the double dual of finite-dimensional vector spaces (a special case, for real and complex vector spaces).

The duality interchanges the subcategories of discrete groups and compact groups. If R is a ring and G is a left R-module, the dual group G^ will become a right R-module; in this way we can also see that discrete left R-modules will be Pontryagin dual to compact right R-modules. The ring End(G) of endomorphisms in LCA is changed by duality into its opposite ring (change the multiplication to the other order). For example if G is an infinite cyclic discrete group, G^ is a circle group: the former has End(G) = Z so this is true also of the latter.