  A geometrisation scheme internal to the category of Lie supergroups is
discussed for the supersymmetric de Rham cocycles on the super-Minkowski group
$\,\mathbb{T}\,$ which determine the standard super-$p$-brane dynamics with
that target, and interpreted within Cartan's approach to the modelling of
orbispaces of group actions by homotopy quotients. The ensuing higher geometric
objects are shown to carry a canonical equivariant structure for the action of
a discrete subgroup of $\,\mathbb{T}$, which results in their descent to the
corresponding orbifolds of $\,\mathbb{T}\,$ and in the emergence of a novel
class of superfield theories with defects.
