  An explicit gerbe-theoretic description of the super-$\sigma$-models of the
Green-Schwarz type is proposed and its fundamental structural properties, such
as equivariance with respect to distinguished isometries of the target
supermanifold and $\kappa$-symmetry, are studied at length for targets with the
structure of a homogeneous space of a Lie supergroup. The programme of
(super)geometrisation of the Cartan-Eilenberg super-$(p+2)$-cocycles that
determine the topological content of the super-$p$-brane mechanics and ensure
its $\kappa$-symmetry, motivated by the successes of and guided by the
intuitions provided by its bosonic predecessor, is based on the idea of a
(super)central extension of a Lie supergroup in the presence of a nontrivial
super-2-cocycle in the Chevalley-Eilenberg cohomology of its Lie superalgebra,
the gap between the two cohomologies being bridged by a super-variant of the
classic Chevalley-Eilenberg construction. A systematic realisation of the
programme is herewith begun with a detailed study of the elementary homogeneous
space of the super-Poincar\'e group, the super-Minkowskian spacetime, whose
simplicity affords straightforward identification of the supergeometric
mechanisms and unobstructed development of formal tools to be employed in more
complex circumstances.
