  We introduce a wide category of superspaces, called locally finitely
generated, which properly includes supermanifolds but enjoys much stronger
permanence properties, as are prompted by applications. Namely, it is closed
under taking finite fibre products (i.e. is finitely complete) and thickenings
by spectra of Weil superalgebras. Nevertheless, in this category, morphisms
with values in a supermanifold are still given in terms of coordinates. This
framework gives a natural notion of relative supermanifolds over a locally
finitely generated base. Moreover, the existence of inner homs, whose source is
the spectrum of a Weil superalgebra, is established; they are generalisations
of the Weil functors defined for smooth manifolds.
