  With a view towards applications in the theory of infinite-dimensional
representations of finite-dimensional Lie supergroups, we introduce a new
category of supermanifolds. In this category, supermanifolds of `maps' and
`fields' (fibre bundle sections) exist. In particular, loop supergroups can be
realised globally in this framework. It also provides a convenient setting for
induced representations of supergroups, allowing for a version of Frobenius
reciprocity. Finally, convolution algebras of finite-dimensional Lie
supergroups are introduced and applied to a prove a supergroup
Dixmier-Malliavin Theorem: The space of smooth vectors of a continuous
representation of a supergroup pair equals the Garding space given by the
convolution with compactly supported smooth supergroup densities.
