  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 consider some applications to T-duality and quantum
Gromov-Hausdorff distance.
