I would like to ask if there is any reference in which Pontrjagin Duality is proved in a categorical context: I started reading the Pontrjagin Dual entry in nLab,
http://ncatlab.org/nlab/show/Pontrjagin+dual;
and I am searching for a proof of the Pontrjagin duality theorem 1 of the link - in the spirit of the paragraph that follows the statement of the result. Simply because I want to have a look at the general theory, which allows one to describe categorically the duals of some categories, than go through all the details needed from the harmonic analysis perspective - which is the case with most of the texts I looked at.
You can put Pontrjagin duality in a nice categorical framework, namely as an adjunction which turns out to be an equivalence (as explained on the nlab), but the proof that the unit of the adjunction i.e. the inclusion to the bidual $G \to \widehat{\widehat{G}}$ is an isomorphism, cannot be done by abstract nonsense. This stays a nontrivial and deep result of harmonic analysis and it is by no means formal (as far as I know).