See also Pontryagin duality.
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
Let $A$ be a commutative (Hausdorff) topological group. A (continuous) group character of $A$ is any continuous homomorphism $\chi: A\to S^1$ to the circle group. The Pontrjagin dual group $\widehat{A}$ is the commutative group of all characters of $A$ with pointwise multiplication (that is multiplication induced by multiplication in the circle group, the multiplication of norm-$1$ complex numbers in $S^1\subset\mathbb{C}$) and with the topology of uniform convergence on each compact $K\subset A$ (this is equivalent to the compact-open topology).
For example, the Pontrjagin dual of the additive group of integers $\mathbb{Z}$ is the circle group $S^1$, and conversely, $\mathbb{Z}$ is the Pontrjagin dual of $S^1$. This pairing of dual topological groups, given by $(n,z) \mapsto z^n$, is related to the subject of Fourier series. In general, the dual of a discrete group is a compact group and conversely. The group $\hat{\mathbb{R}}$ is isomorphic again to $\mathbb{R}$ (the additive group of real numbers), with the pairing given by $(x,p) \mapsto \mathrm{e}^{\mathrm{i} x p}$; similarly, $\hat{\mathbb{R}^n}$ is isomorphic to the Cartesian space $\mathbb{R}^n$.
For every locally compact (Hausdorff) topological abelian group $A$, the natural function $A \mapsto \widehat{\widehat{A}}$ from $A$ into the Pontrjagin dual of the Pontrjagin dual of $A$, assigning to every $g\in A$ the continuous character $f_g$ given by $f_g(\chi)=\chi(g)$, is an isomorphism of topological groups (that is, a group isomorphism that is also a homeomorphism).
Thus, the functor
is an equivalence of categories, in fact an adjoint equivalence whose unit is
and whose counit (the same arrow read in the opposite category) are isomorphisms. This contravariant self-equivalence restricts to equivalences
[]
where $Ab$ is the category of (discrete topological) groups and $CompAb$ is the category of compact Hausdorff topological abelian groups, each embedded in $LocCompAb$ in the evident way.
The Fourier transform on locally compact abelian groups is formulated in terms of Pontrjagin duals (see below).
Also see:
which provides a perhaps better context for Pontryagin duality than the category of locally compact Hausdorff abelian groups (also known as ‘LCA groups’). Barr explains:
Did you know that there is a *-autonomous category of topological abelian groups that includes all the LCA groups and whose duality extends that of Pontrjagin? The groups are characterized by the property that among all topological groups on the same underlying abelian group and with the same set of continuous homomorphisms to the circle, these have the finest topology. It is not obvious that such a finest exists, but it does and that is the key.
There are many properties of locally compact Hausdorff abelian groups that implies properties of their Pontrjagin duals. For example:
If $A$ is finite, $\widehat{A}$ is finite.
If $A$ is compact, $\widehat{A}$ is discrete.
If $A$ is discrete, $\widehat{A}$ is compact.
If $A$ is torsion-free and discrete, $\widehat{A}$ is connected and compact.
If $A$ is an abelian torsion group then $\widehat A$ is an abelian profinite group (for more see at Pontryagin duality for torsion abelian groups)
If $A$ is connected and compact, $\widehat{A}$ is torsion-free and discrete.
If $A$ is a Lie group, $\widehat{A}$ has finite rank.
If A has finite rank, $\widehat{A}$ is a Lie group.
If $A$ is second countable, $\widehat{A}$ is second countable.
If $A$ is separable, $\widehat{A}$ is metrizable.
For a discussion of these facts, with some references, try:
Variations on Pontryagin duality, (nCafe)
Sidney A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Soc. Lecture Notes 29, Cambridge U. Press, 1977.
and this more advanced text:
Pontrjagin duality underlies the abstract framework of Fourier analysis on locally compact Hausdorff abelian groups $A$: by Fourier duality? on $A$, there is a Hilbert space isomorphism (Fourier transform)
where $d\mu$ is a suitable choice of Haar measure on $A$, and $d\hat{\mu}$ is a suitable choice of Haar measure on the dual group. Fourier duality is compatible with Pontrjagin duality in the sense that if $\hat{\hat{A}}$ is identified with $A$, then $\mathcal{F}_{\hat{A}}$ is the inverse of $\mathcal{F}_A$.
There is a recent categorification of the Pontrjagin duality theorem, motivated by applications to topological T-duality: