Metric Tannakian duality

We incorporate metric data into the framework of Tannaka–Krein duality. Thus, for any group with left invariant metric, we produce a dual metric on its category of unitary representations. We characterize the conditions under which a “double-dual” metric on the group may be recovered from the metric on representations, and provide conditions under which a metric agrees with its double-dual. We also explore a diverse class of possible applications of the theory, including applications to TT-duality and to quantum Gromov–Hausdorff distance.