Pontrjagin Duality

A $\ast$-autonomous category of topological abelian groups extending Pontrjagin duality

Let $\mathcal{G}$ denote the category of topological abelian groups that can be embedded topologically and algebraically into a possibly infinite product of locally compact abelian groups, with continuous homomorphisms as morphisms.

There is a full subcategory $\mathcal{S} \subset \mathcal{G}$ s.t.

  1. $\mathcal{S}$ is complete and cocomplete;
  2. $\mathcal{S}$ is $\ast$-autonomous.

The $\ast$-autonomous category structure is given by:

  1. The global dualizing object $\bot:= \mathbb{T}$, the circle group.
  2. The dualization functor $(\cdot)^\ast := (\cdot) \multimap \bot$ where $A\multimap B := \operatorname{Hom}(A,B)$ is the internal hom, equipped with a topology that makes it into an object of $\mathcal{S}$.
  3. The monoidal structure $\otimes$ which is the tensor product $|A|\otimes |B|$ where $|G|$ is the underlying discrete group of $G$.

See M. Barr, On duality of topological abelian groups or Barr, Kleisi. On Mackey Topologies In Topological Abelian Groups