From Recent Advances in the Langlands Program, quoted in This Week’s Finds:

First of all, it should be remarked that according to the Tannakian phylosophy, one can reconstruct a group from the category of its finite-dimensional representations, equipped with the structure of the tensor product.

I suppose one should think of this as the categorification of Pontrjagin duality?

For a long while, I had wondered how this goes; this Introduction to Tannaka Duality and Quantum Groups will probably answer my questions.