  We reformulate superalgebra and supergeometry in completely categorical terms
by a consequent use of the functor of points. The increased abstraction of this
approach is rewarded by a number of great advantages. First, we show that one
can extend supergeometry completely naturally to infinite-dimensional contexts.
Secondly, some subtle and sometimes obscure-seeming points of supergeometry
become clear in light of these results, e.g., the relation between the
Berezin-Leites-Kostant and the de Witt-Rogers approaches, and the precise
geometric meaning of odd parameters in supergeometry. In addition, this method
allows us to construct in an easy manner superspaces of morphisms between
superobjects, i.e., the inner Hom objects associated with the sets of such
morphisms. The results of our work rely heavily and were inspired by the ideas
of V. Molotkov, who first outlined the approach extensively expounded here.
