  The theme here is category-measure duality, in the context of a topological
group. One can often handle the (Baire) category case and the (Lebesgue, or
Haar) measure cases together, by working bi-topologically: switching between
the original topology and a suitable refinement (a density topology). This
prompts a systematic study of such density topologies, and the corresponding
$\sigma$-ideals of negligibles. Such ideas go back to Weil's classic book, and
to Hashimoto's ideal topologies. We make use of group norms, which cast light
on the interplay between the group and measure structures. The Steinhaus-Weil
interior-points theorem (`on $AA^{-1}$') plays a crucial role here; so too does
its converse, the Simmons-Mospan theorem.
